K11n115

K11n114

K11n116

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n115 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X14,6,15,5 X7,19,8,18 X9,17,10,16 X2,11,3,12 X20,13,21,14 X22,16,1,15 X17,9,18,8 X12,19,13,20 X6,21,7,22
Gauss code	1, -6, 2, -1, 3, -11, -4, 9, -5, -2, 6, -10, 7, -3, 8, 5, -9, 4, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 10 14 -18 -16 2 20 22 -8 12 6

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n115/ThurstonBennequinNumber Hyperbolic Volume 14.7634 A-Polynomial See Data:K11n115/A-polynomial

[edit Notes for K11n115's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for K11n115's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+6t^{2}-18t+27-18t^{-1}+6t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 77, 0 }
Jones polynomial	$2q^{4}-5q^{3}+9q^{2}-12q+13-13q^{-1}+11q^{-2}-7q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+2a^{2}z^{4}+z^{4}a^{-2}-3z^{4}-a^{4}z^{2}+3a^{2}z^{2}-5z^{2}+2a^{2}+a^{-4}-2$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+4a^{2}z^{8}+2z^{8}a^{-2}+6z^{8}+6a^{3}z^{7}+11az^{7}+6z^{7}a^{-1}+z^{7}a^{-3}+4a^{4}z^{6}+2a^{2}z^{6}+z^{6}a^{-2}-z^{6}+a^{5}z^{5}-10a^{3}z^{5}-24az^{5}-10z^{5}a^{-1}+3z^{5}a^{-3}-7a^{4}z^{4}-17a^{2}z^{4}-z^{4}a^{-2}+3z^{4}a^{-4}-14z^{4}-a^{5}z^{3}+4a^{3}z^{3}+15az^{3}+6z^{3}a^{-1}-4z^{3}a^{-3}+3a^{4}z^{2}+12a^{2}z^{2}-z^{2}a^{-2}-4z^{2}a^{-4}+12z^{2}-a^{3}z-4az-2za^{-1}+za^{-3}-2a^{2}+a^{-4}-2$
The A2 invariant	Data:K11n115/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n115/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n115 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11n115"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-t^{3}+6t^{2}-18t+27-18t^{-1}+6t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{6}-3z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 77, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$2q^{4}-5q^{3}+9q^{2}-12q+13-13q^{-1}+11q^{-2}-7q^{-3}+4q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-z^{6}+2a^{2}z^{4}+z^{4}a^{-2}-3z^{4}-a^{4}z^{2}+3a^{2}z^{2}-5z^{2}+2a^{2}+a^{-4}-2$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$az^{9}+z^{9}a^{-1}+4a^{2}z^{8}+2z^{8}a^{-2}+6z^{8}+6a^{3}z^{7}+11az^{7}+6z^{7}a^{-1}+z^{7}a^{-3}+4a^{4}z^{6}+2a^{2}z^{6}+z^{6}a^{-2}-z^{6}+a^{5}z^{5}-10a^{3}z^{5}-24az^{5}-10z^{5}a^{-1}+3z^{5}a^{-3}-7a^{4}z^{4}-17a^{2}z^{4}-z^{4}a^{-2}+3z^{4}a^{-4}-14z^{4}-a^{5}z^{3}+4a^{3}z^{3}+15az^{3}+6z^{3}a^{-1}-4z^{3}a^{-3}+3a^{4}z^{2}+12a^{2}z^{2}-z^{2}a^{-2}-4z^{2}a^{-4}+12z^{2}-a^{3}z-4az-2za^{-1}+za^{-3}-2a^{2}+a^{-4}-2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11n115"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $-t^{3}+6t^{2}-18t+27-18t^{-1}+6t^{-2}-t^{-3}$, $2q^{4}-5q^{3}+9q^{2}-12q+13-13q^{-1}+11q^{-2}-7q^{-3}+4q^{-4}-q^{-5}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-3, -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 $-12$ $-8$ $72$ $98$ $38$ $96$ ${\frac {496}{3}}$ ${\frac {160}{3}}$ $24$ $-288$ $32$ $-1176$ $-456$ $-{\frac {10031}{10}}$ ${\frac {3026}{15}}$ $-{\frac {13862}{15}}$ ${\frac {751}{6}}$ $-{\frac {2191}{10}}$

V_2,1 through V_6,9 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 $t^{r}q^{j}$ 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=$0 is the signature of K11n115. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                   2 2 7                 3   -3 5               6 2   4 3             6 3     -3 1           7 6       1 -1         7 7         0 -3       4 6           -2 -5     3 7             4 -7   1 4               -3 -9   3                 3 -11 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$

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). Back to the top.

K11n114 K11n116