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Number the seven crossings from top to bottom. Orient the strand entering the top-left arch downward. With the usual positive braid crossing convention, the crossing signs areTo make the diagram calculation reproducible, its oriented Gauss code isSubscripts record overpassing and underpassing. In this convention a positive trefoil knot has knot signature $-2$.

Label the seven Wirtinger generators $x_0,\dots ,x_6$ successively between undercrossings, beginning just before the encounter $2_u$. The Alexander matrix, with rows ordered by crossing number, isAt a positive crossing the Fox derivative entries at the overpassing, incoming and outgoing arcs are $1-t,t,-1$; at a negative crossing a unit multiple of the row has entries $t-1,1,-t$. Deleting the last row and column gives determinant $t(t^4-5t^3+7t^2-5t+1)$. Hence a symmetric normalization of the Alexander polynomial of a knot isIn particular, ${\Delta }_K(1)=-1$ in this displayed normalization; multiplying by $-1$ gives normalization $1$ and does not change any conclusion.

The Seifert algorithm produces four Seifert circles and seven bands. Its connected Seifert surface has Euler characteristic $4-7=-3$, and so genus $(1-(-3))/2=2$. Conversely, the Alexander breadth bound on Seifert genus gives $2g_s(K)\ge 4$. ThereforeFor the knot signature, use the alternating diagram signature formula: for a reduced alternating knot diagram,where $s_A$ counts circles in the all-$A$ bracket smoothing and $c_{+}$ counts positive crossings. Here the all-$A$ smoothing has four circles, the all-$B$ smoothing has five, and $c_{+}=5$. ThusThe opposite global knot signature convention gives $+2$ instead.

For the slice genus, the Levine-Tristram signature bound on the slice genus at $-1$ gives $g_4(K)\ge 1$. There is also an explicit unknotting crossing: switch crossing $3$. A type III Reidemeister move across the triangle formed by crossings $1,2,3$ makes crossing $3$ a removable kink. Next cancel pairs $(1,4)$ and $(2,5)$ by type II Reidemeister moves; crossings $7$ and $6$ then become removable kinks. This leaves the unknot. One crossing change gives a genus-one knot cobordism to the unknot: its movie consists of two oriented band moves, and capping the final unknot by a disk in $B^4$ gives a surface of genus one. Consequently

There are two common normalizations. I will define both explicitly, because the numerical conclusion in the question uses the unreduced one.

Set $\delta =-A^2-A^{-2}$. The reduced Kauffman bracket of a nonempty link diagram iswhere $s$ runs over all bracket smoothings, $a(s),b(s)$ count $A$- and $B$-smoothings, and $|s|$ is the number of resulting circles. Locally,Fix the usual Kauffman bracket convention in which the $A$-smoothing at a positive braid crossing is its oriented smoothing; reflecting a crossing exchanges $A$ and $A^{-1}$. The unknot has reduced bracket $1$, and adjoining another circle multiplies the bracket by $\delta$. The unreduced Kauffman bracket uses ${\delta }^{|s|}$ instead and assigns the empty diagram $1$. Thus for a nonempty diagram

For completeness, the bracket calculations for all three Reidemeister moves can be carried out in the Temperley-Lieb diagram algebra. Let $I$ be the identity two-strand tangle, and let $e$ be the cap-cup tangle. Composition gives $e^2=\delta e$. The two crossings correspond toConsequentlywhich proves invariance under the second Reidemeister move. On three strands, the cap-cup diagrams satisfy $e_1{e}_2{e}_1=e_1$, $e_2{e}_1{e}_2=e_2$, and $e_i^2=\delta e_i$. Expand the two sides of the third move. Their difference isThus the third Reidemeister move also preserves either bracket normalization. Reflected forms of these moves follow by replacing $A$ with $A^{-1}$.

A positive curl contributes $A\delta +A^{-1}=-A^3$, and a negative curl contributes $A^{-1}\delta +A=-A^{-3}$. The writhe of a link diagram $w(D)$ changes by $+1$ or $-1$ in exactly these cases. Therefore the corrected bracketis invariant under the first Reidemeister move as well; the second and third moves leave the writhe of a link diagram unchanged. Substituting $t=A^{-4}$ defines a Jones polynomial. For multiple components half-integer powers of $t$ may occur. The reduced version $V_L^{\mathrm{red}}$ has $V_U^{\mathrm{red}}=1$, and the Unreduced Jones polynomial is

At $A=1$, switching a crossing does not change the bracket: both smoothing coefficients are $1$. It changes the writhe of a link diagram by $2$, so the correction factor $(-1{)}^{-w}$ is unchanged. Hence either Jones polynomial at $t=1$ is unchanged by a crossing change. By switching crossings, any $l$-component link can be made an unlink. Its crossing-free diagram has $l$ circles and zero writhe of a link diagram, givingThus the printed conclusion $|V_L(1)|=2^l$ holds for the unreduced normalization. For the reduced Jones polynomial, the correct conclusion is $\boxed{|V_L^{\mathrm{red}}(1)|=2^{l-1}}$; the unknot alone already rules out $2^l$ in that convention.

An unknotting crossing is a crossing whose crossing change converts a knot into the unknot. It gives a genus-one knot cobordism to the unknot, and hence slice genus at most one after capping with a disk in $B^4$.