# K11n71

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n71 at Knotilus!

### Knot presentations

 Planar diagram presentation X4251 X8493 X14,6,15,5 X2837 X9,20,10,21 X16,12,17,11 X6,14,7,13 X18,16,19,15 X12,18,13,17 X19,22,20,1 X21,10,22,11 Gauss code 1, -4, 2, -1, 3, -7, 4, -2, -5, 11, 6, -9, 7, -3, 8, -6, 9, -8, -10, 5, -11, 10 Dowker-Thistlethwaite code 4 8 14 2 -20 16 6 18 12 -22 -10
A Braid Representative
A Morse Link Presentation

### Three dimensional invariants

 Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 4 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n71/ThurstonBennequinNumber Hyperbolic Volume 13.5931 A-Polynomial See Data:K11n71/A-polynomial

### Four dimensional invariants

 Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $1$ Rasmussen s-Invariant -2

### Polynomial invariants

 Alexander polynomial $2 t^3-7 t^2+14 t-17+14 t^{-1} -7 t^{-2} +2 t^{-3}$ Conway polynomial $2 z^6+5 z^4+4 z^2+1$ 2nd Alexander ideal (db, data sources) $\left\{t^2-t+1\right\}$ Determinant and Signature { 63, 2 } Jones polynomial $q^9-3 q^8+5 q^7-9 q^6+10 q^5-10 q^4+11 q^3-7 q^2+5 q-2$ HOMFLY-PT polynomial (db, data sources) $2 z^6 a^{-4} -2 z^4 a^{-2} +10 z^4 a^{-4} -3 z^4 a^{-6} -5 z^2 a^{-2} +18 z^2 a^{-4} -10 z^2 a^{-6} +z^2 a^{-8} -3 a^{-2} +11 a^{-4} -9 a^{-6} +2 a^{-8}$ Kauffman polynomial (db, data sources) $z^9 a^{-5} +z^9 a^{-7} +4 z^8 a^{-4} +7 z^8 a^{-6} +3 z^8 a^{-8} +4 z^7 a^{-3} +8 z^7 a^{-5} +7 z^7 a^{-7} +3 z^7 a^{-9} +z^6 a^{-2} -10 z^6 a^{-4} -18 z^6 a^{-6} -6 z^6 a^{-8} +z^6 a^{-10} -10 z^5 a^{-3} -31 z^5 a^{-5} -31 z^5 a^{-7} -10 z^5 a^{-9} +4 z^4 a^{-2} +19 z^4 a^{-4} +17 z^4 a^{-6} -z^4 a^{-8} -3 z^4 a^{-10} +3 z^3 a^{-1} +19 z^3 a^{-3} +42 z^3 a^{-5} +36 z^3 a^{-7} +10 z^3 a^{-9} -6 z^2 a^{-2} -20 z^2 a^{-4} -15 z^2 a^{-6} +z^2 a^{-8} +2 z^2 a^{-10} -3 z a^{-1} -11 z a^{-3} -21 z a^{-5} -17 z a^{-7} -4 z a^{-9} +3 a^{-2} +11 a^{-4} +9 a^{-6} +2 a^{-8}$ The A2 invariant Data:K11n71/QuantumInvariant/A2/1,0 The G2 invariant Data:K11n71/QuantumInvariant/G2/1,0

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_65, 10_77, K11n75,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n75,}

### Vassiliev invariants

 V2 and V3: (4, 5)
V2,1 through V6,9:
 V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $16$ $40$ $128$ $\frac{536}{3}$ $\frac{64}{3}$ $640$ $\frac{2416}{3}$ $\frac{352}{3}$ $104$ $\frac{2048}{3}$ $800$ $\frac{8576}{3}$ $\frac{1024}{3}$ $\frac{58622}{15}$ $\frac{3632}{15}$ $\frac{53888}{45}$ $\frac{370}{9}$ $\frac{1982}{15}$

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

### Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$2 is the signature of K11n71. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-1012345678χ
19         11
17        2 -2
15       31 2
13      62  -4
11     43   1
9    66    0
7   54     1
5  26      4
3 35       -2
1 3        3
-12         -2
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-1$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=8$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.