K11n92

K11n91

K11n93

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n92 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X7,20,8,21 X9,16,10,17 X2,11,3,12 X13,19,14,18 X15,8,16,9 X17,1,18,22 X19,6,20,7 X21,15,22,14
Gauss code	1, -6, 2, -1, 3, 10, -4, 8, -5, -2, 6, -3, -7, 11, -8, 5, -9, 7, -10, 4, -11, 9
Dowker-Thistlethwaite code	4 10 12 -20 -16 2 -18 -8 -22 -6 -14

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:K11n92/ThurstonBennequinNumber Hyperbolic Volume 8.77077 A-Polynomial See Data:K11n92/A-polynomial

[edit Notes for K11n92'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 K11n92's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-3t^{2}+3t-1+3t^{-1}-3t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+3z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 15, -2 }
Jones polynomial	$-q^{4}+2q^{3}-2q^{2}+3q-2+2q^{-1}-2q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}-a^{2}z^{4}-z^{4}a^{-2}+5z^{4}-4a^{2}z^{2}-3z^{2}a^{-2}+7z^{2}+a^{4}-3a^{2}-a^{-2}+4$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+a^{2}z^{8}+2z^{8}a^{-2}+3z^{8}-5az^{7}-4z^{7}a^{-1}+z^{7}a^{-3}-6a^{2}z^{6}-11z^{6}a^{-2}-17z^{6}+6az^{5}+z^{5}a^{-1}-5z^{5}a^{-3}+11a^{2}z^{4}+17z^{4}a^{-2}+28z^{4}-a^{3}z^{3}-3az^{3}+4z^{3}a^{-1}+6z^{3}a^{-3}-a^{4}z^{2}-10a^{2}z^{2}-9z^{2}a^{-2}-18z^{2}+a^{3}z+az-za^{-1}-za^{-3}+a^{4}+3a^{2}+a^{-2}+4$
The A2 invariant	$q^{14}-q^{8}-q^{6}+2+q^{-2}+q^{-4}+q^{-6}-q^{-12}$
The G2 invariant	Data:K11n92/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n92 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["K11n92"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-3t^{2}+3t-1+3t^{-1}-3t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+3z^{4}+1$

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]=

{ 15, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{4}+2q^{3}-2q^{2}+3q-2+2q^{-1}-2q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}-a^{2}z^{4}-z^{4}a^{-2}+5z^{4}-4a^{2}z^{2}-3z^{2}a^{-2}+7z^{2}+a^{4}-3a^{2}-a^{-2}+4$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{9}+z^{9}a^{-1}+a^{2}z^{8}+2z^{8}a^{-2}+3z^{8}-5az^{7}-4z^{7}a^{-1}+z^{7}a^{-3}-6a^{2}z^{6}-11z^{6}a^{-2}-17z^{6}+6az^{5}+z^{5}a^{-1}-5z^{5}a^{-3}+11a^{2}z^{4}+17z^{4}a^{-2}+28z^{4}-a^{3}z^{3}-3az^{3}+4z^{3}a^{-1}+6z^{3}a^{-3}-a^{4}z^{2}-10a^{2}z^{2}-9z^{2}a^{-2}-18z^{2}+a^{3}z+az-za^{-1}-za^{-3}+a^{4}+3a^{2}+a^{-2}+4$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {10_136,}

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["K11n92"];

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]=

{ $t^{3}-3t^{2}+3t-1+3t^{-1}-3t^{-2}+t^{-3}$, $-q^{4}+2q^{3}-2q^{2}+3q-2+2q^{-1}-2q^{-2}+q^{-3}$ }

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]=

{10_136,}

Vassiliev invariants

V2 and V3:

(0, 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 $0$ $8$ $0$ $-32$ $-24$ $0$ ${\frac {80}{3}}$ $-{\frac {64}{3}}$ $40$ $0$ $32$ $0$ $0$ $32$ ${\frac {40}{3}}$ $40$ $-48$ $0$

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 $s=$-2 is the signature of K11n92. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

Computer Talk

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

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.

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