K11n15

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 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n15 at Knotilus!

Knot presentations

 Planar diagram presentation X4251 X8394 X10,6,11,5 X7,16,8,17 X2,9,3,10 X11,19,12,18 X13,21,14,20 X15,6,16,7 X17,1,18,22 X19,13,20,12 X21,15,22,14 Gauss code 1, -5, 2, -1, 3, 8, -4, -2, 5, -3, -6, 10, -7, 11, -8, 4, -9, 6, -10, 7, -11, 9 Dowker-Thistlethwaite code 4 8 10 -16 2 -18 -20 -6 -22 -12 -14

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:K11n15/ThurstonBennequinNumber Hyperbolic Volume 9.94805 A-Polynomial See Data:K11n15/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 ${\displaystyle -q^{8}+2q^{7}-3q^{6}+5q^{5}-6q^{4}+6q^{3}-5q^{2}+4q-2+q^{-1}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle z^{6}a^{-4}-2z^{4}a^{-2}+5z^{4}a^{-4}-z^{4}a^{-6}-6z^{2}a^{-2}+9z^{2}a^{-4}-3z^{2}a^{-6}+z^{2}-4a^{-2}+5a^{-4}-2a^{-6}+2}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{9}a^{-3}+z^{9}a^{-5}+z^{8}a^{-2}+3z^{8}a^{-4}+2z^{8}a^{-6}-5z^{7}a^{-3}-3z^{7}a^{-5}+2z^{7}a^{-7}-5z^{6}a^{-2}-14z^{6}a^{-4}-7z^{6}a^{-6}+2z^{6}a^{-8}+2z^{5}a^{-1}+12z^{5}a^{-3}+4z^{5}a^{-5}-5z^{5}a^{-7}+z^{5}a^{-9}+14z^{4}a^{-2}+29z^{4}a^{-4}+10z^{4}a^{-6}-6z^{4}a^{-8}+z^{4}-4z^{3}a^{-1}-8z^{3}a^{-3}+z^{3}a^{-5}+2z^{3}a^{-7}-3z^{3}a^{-9}-15z^{2}a^{-2}-21z^{2}a^{-4}-6z^{2}a^{-6}+3z^{2}a^{-8}-3z^{2}+za^{-1}+za^{-3}-za^{-5}+za^{-9}+4a^{-2}+5a^{-4}+2a^{-6}+2}$ The A2 invariant ${\displaystyle q^{4}+q^{2}+q^{-2}-2q^{-4}+2q^{-12}+2q^{-16}-q^{-20}-q^{-24}}$ The G2 invariant Data:K11n15/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {9_12,}

Vassiliev invariants

 V2 and V3: (1, 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 ${\displaystyle 4}$ ${\displaystyle 24}$ ${\displaystyle 8}$ ${\displaystyle {\frac {158}{3}}}$ ${\displaystyle -{\frac {38}{3}}}$ ${\displaystyle 96}$ ${\displaystyle 144}$ ${\displaystyle -32}$ ${\displaystyle -8}$ ${\displaystyle {\frac {32}{3}}}$ ${\displaystyle 288}$ ${\displaystyle {\frac {632}{3}}}$ ${\displaystyle -{\frac {152}{3}}}$ ${\displaystyle {\frac {20911}{30}}}$ ${\displaystyle {\frac {286}{5}}}$ ${\displaystyle {\frac {422}{45}}}$ ${\displaystyle -{\frac {175}{18}}}$ ${\displaystyle -{\frac {689}{30}}}$

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 K11n15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-2-101234567χ
17         1-1
15        1 1
13       21 -1
11      31  2
9     32   -1
7    33    0
5   23     1
3  23      -1
1 13       2
-1 1        -1
-31         1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=1}$ ${\displaystyle i=3}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

Computer Talk

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