# K11n58

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n58 at Knotilus!

### Knot presentations

 Planar diagram presentation X4251 X8493 X14,5,15,6 X2837 X9,17,10,16 X11,18,12,19 X13,20,14,21 X6,15,7,16 X17,1,18,22 X19,12,20,13 X21,10,22,11 Gauss code 1, -4, 2, -1, 3, -8, 4, -2, -5, 11, -6, 10, -7, -3, 8, 5, -9, 6, -10, 7, -11, 9 Dowker-Thistlethwaite code 4 8 14 2 -16 -18 -20 6 -22 -12 -10

### Three dimensional invariants

 Symmetry type Reversible Unknotting number ${\displaystyle \{1,2\}}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n58/ThurstonBennequinNumber Hyperbolic Volume 9.97833 A-Polynomial See Data:K11n58/A-polynomial

### Four dimensional invariants

 Smooth 4 genus Missing Topological 4 genus Missing Concordance genus ${\displaystyle 3}$ Rasmussen s-Invariant 2

### Polynomial invariants

 Alexander polynomial ${\displaystyle t^{3}-4t^{2}+8t-9+8t^{-1}-4t^{-2}+t^{-3}}$ Conway polynomial ${\displaystyle z^{6}+2z^{4}+z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 35, -2 } Jones polynomial ${\displaystyle -q^{4}+2q^{3}-3q^{2}+5q-5+6q^{-1}-5q^{-2}+4q^{-3}-3q^{-4}+q^{-5}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle z^{6}-2a^{2}z^{4}-z^{4}a^{-2}+5z^{4}+a^{4}z^{2}-6a^{2}z^{2}-3z^{2}a^{-2}+9z^{2}+a^{4}-4a^{2}-2a^{-2}+6}$ Kauffman polynomial (db, data sources) ${\displaystyle az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+2z^{8}a^{-2}+4z^{8}+2a^{3}z^{7}-az^{7}-2z^{7}a^{-1}+z^{7}a^{-3}+a^{4}z^{6}-6a^{2}z^{6}-10z^{6}a^{-2}-17z^{6}-5a^{3}z^{5}-5az^{5}-5z^{5}a^{-1}-5z^{5}a^{-3}+8a^{2}z^{4}+15z^{4}a^{-2}+23z^{4}+3a^{5}z^{3}+6a^{3}z^{3}+6az^{3}+10z^{3}a^{-1}+7z^{3}a^{-3}+a^{6}z^{2}-10a^{2}z^{2}-8z^{2}a^{-2}-17z^{2}-a^{5}z-3a^{3}z-4az-4za^{-1}-2za^{-3}+a^{4}+4a^{2}+2a^{-2}+6}$ The A2 invariant ${\displaystyle q^{16}-q^{12}-2q^{8}+q^{2}+3+q^{-2}+2q^{-4}-q^{-8}-q^{-12}}$ The G2 invariant Data:K11n58/QuantumInvariant/G2/1,0

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

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {K11n56,}

### Vassiliev invariants

 V2 and V3: (1, 1)
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 ${\displaystyle 4}$ ${\displaystyle 8}$ ${\displaystyle 8}$ ${\displaystyle -{\frac {34}{3}}}$ ${\displaystyle -{\frac {38}{3}}}$ ${\displaystyle 32}$ ${\displaystyle {\frac {176}{3}}}$ ${\displaystyle -{\frac {64}{3}}}$ ${\displaystyle 40}$ ${\displaystyle {\frac {32}{3}}}$ ${\displaystyle 32}$ ${\displaystyle -{\frac {136}{3}}}$ ${\displaystyle -{\frac {152}{3}}}$ ${\displaystyle -{\frac {209}{30}}}$ ${\displaystyle {\frac {1258}{15}}}$ ${\displaystyle -{\frac {3898}{45}}}$ ${\displaystyle -{\frac {463}{18}}}$ ${\displaystyle -{\frac {1169}{30}}}$

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 ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$). The squares with yellow highlighting are those on the "critical diagonals", where ${\displaystyle j-2r=s+1}$ or ${\displaystyle j-2r=s-1}$, where ${\displaystyle s=}$-2 is the signature of K11n58. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-4-3-2-1012345χ
9         1-1
7        1 1
5       21 -1
3      31  2
1     22   0
-1    43    1
-3   23     1
-5  23      -1
-7 12       1
-9 2        -2
-111         1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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