# K11n6

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 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n6 at Knotilus!

 Knot K11n6. A graph, knot K11n6. A part of a knot and a part of a graph.

### Knot presentations

 Planar diagram presentation X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X11,18,12,19 X6,14,7,13 X15,20,16,21 X17,12,18,13 X19,22,20,1 X21,16,22,17 Gauss code 1, -5, 2, -1, 3, -7, 4, -2, 5, -3, -6, 9, 7, -4, -8, 11, -9, 6, -10, 8, -11, 10 Dowker-Thistlethwaite code 4 8 10 14 2 -18 6 -20 -12 -22 -16
A Braid Representative
A Morse Link Presentation

### Three dimensional invariants

 Symmetry type Chiral Unknotting number ${\displaystyle \{1,2\}}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n6/ThurstonBennequinNumber Hyperbolic Volume 11.407 A-Polynomial See Data:K11n6/A-polynomial

### Four dimensional invariants

 Smooth 4 genus Missing Topological 4 genus Missing Concordance genus ${\displaystyle 3}$ Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial ${\displaystyle -t^{3}+3t^{2}-3t+3-3t^{-1}+3t^{-2}-t^{-3}}$ Conway polynomial ${\displaystyle -z^{6}-3z^{4}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 17, 0 } Jones polynomial ${\displaystyle q^{3}-q^{2}+1-2q^{-1}+3q^{-2}-3q^{-3}+4q^{-4}-3q^{-5}+2q^{-6}-q^{-7}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -z^{2}a^{6}-2a^{6}+2z^{4}a^{4}+7z^{2}a^{4}+5a^{4}-z^{6}a^{2}-5z^{4}a^{2}-7z^{2}a^{2}-3a^{2}+z^{2}a^{-2}+a^{-2}}$ Kauffman polynomial (db, data sources) ${\displaystyle a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+4a^{4}z^{8}+2a^{2}z^{8}+a^{7}z^{7}-2a^{5}z^{7}-3a^{3}z^{7}+az^{7}+z^{7}a^{-1}-10a^{6}z^{6}-21a^{4}z^{6}-12a^{2}z^{6}+z^{6}a^{-2}-5a^{7}z^{5}-8a^{5}z^{5}-5a^{3}z^{5}-8az^{5}-6z^{5}a^{-1}+14a^{6}z^{4}+31a^{4}z^{4}+19a^{2}z^{4}-5z^{4}a^{-2}-3z^{4}+7a^{7}z^{3}+15a^{5}z^{3}+13a^{3}z^{3}+13az^{3}+8z^{3}a^{-1}-8a^{6}z^{2}-19a^{4}z^{2}-12a^{2}z^{2}+5z^{2}a^{-2}+4z^{2}-3a^{7}z-6a^{5}z-6a^{3}z-6az-3za^{-1}+2a^{6}+5a^{4}+3a^{2}-a^{-2}}$ The A2 invariant Data:K11n6/QuantumInvariant/A2/1,0 The G2 invariant Data:K11n6/QuantumInvariant/G2/1,0

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

Same Alexander/Conway Polynomial: {8_2,}

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

### Vassiliev invariants

 V2 and V3: (0, -3)
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 ${\displaystyle 0}$ ${\displaystyle -24}$ ${\displaystyle 0}$ ${\displaystyle 80}$ ${\displaystyle 24}$ ${\displaystyle 0}$ ${\displaystyle -112}$ ${\displaystyle 0}$ ${\displaystyle -24}$ ${\displaystyle 0}$ ${\displaystyle 288}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 632}$ ${\displaystyle -{\frac {40}{3}}}$ ${\displaystyle 200}$ ${\displaystyle 56}$ ${\displaystyle 8}$

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=}$0 is the signature of K11n6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-7-6-5-4-3-2-101234χ
7           11
5            0
3        111 -1
1       21   1
-1      221   -1
-3     221    1
-5    22      0
-7   221      1
-9  12        1
-11 12         -1
-13 1          1
-151           -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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

### Modifying This Page

Read me first: Modifying Knot Pages.

See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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