K11a95

K11a94

K11a96

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a95 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X12,6,13,5 X18,8,19,7 X2,10,3,9 X8,12,9,11 X22,14,1,13 X20,16,21,15 X6,18,7,17 X16,20,17,19 X14,22,15,21
Gauss code	1, -5, 2, -1, 3, -9, 4, -6, 5, -2, 6, -3, 7, -11, 8, -10, 9, -4, 10, -8, 11, -7
Dowker-Thistlethwaite code	4 10 12 18 2 8 22 20 6 16 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a95/ThurstonBennequinNumber Hyperbolic Volume 11.7607 A-Polynomial See Data:K11a95/A-polynomial

[edit Notes for K11a95's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $2$ Rasmussen s-Invariant -4

[edit Notes for K11a95's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$6 t^2-18 t+25-18 t^{-1} +6 t^{-2}$
Conway polynomial	$6 z^4+6 z^2+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 73, 4 }
Jones polynomial	$-q^{13}+3 q^{12}-5 q^{11}+7 q^{10}-10 q^9+11 q^8-11 q^7+10 q^6-7 q^5+5 q^4-2 q^3+q^2$
HOMFLY-PT polynomial (db, data sources)	$z^4 a^{-4} +2 z^4 a^{-6} +2 z^4 a^{-8} +z^4 a^{-10} +2 z^2 a^{-4} +3 z^2 a^{-6} +2 z^2 a^{-8} -z^2 a^{-12} + a^{-4} + a^{-6} - a^{-10}$
Kauffman polynomial (db, data sources)	$z^{10} a^{-10} +z^{10} a^{-12} +2 z^9 a^{-9} +5 z^9 a^{-11} +3 z^9 a^{-13} +3 z^8 a^{-8} +2 z^8 a^{-10} +2 z^8 a^{-12} +3 z^8 a^{-14} +3 z^7 a^{-7} -15 z^7 a^{-11} -11 z^7 a^{-13} +z^7 a^{-15} +3 z^6 a^{-6} -3 z^6 a^{-8} -10 z^6 a^{-10} -17 z^6 a^{-12} -13 z^6 a^{-14} +2 z^5 a^{-5} -z^5 a^{-7} -6 z^5 a^{-9} +10 z^5 a^{-11} +9 z^5 a^{-13} -4 z^5 a^{-15} +z^4 a^{-4} -3 z^4 a^{-6} +2 z^4 a^{-8} +9 z^4 a^{-10} +18 z^4 a^{-12} +15 z^4 a^{-14} -2 z^3 a^{-5} -2 z^3 a^{-7} +5 z^3 a^{-9} +z^3 a^{-11} +4 z^3 a^{-15} -2 z^2 a^{-4} +2 z^2 a^{-6} -z^2 a^{-8} -5 z^2 a^{-10} -4 z^2 a^{-12} -4 z^2 a^{-14} +2 z a^{-7} -z a^{-9} -3 z a^{-11} -z a^{-13} -z a^{-15} + a^{-4} - a^{-6} + a^{-10}$
The A2 invariant	Data:K11a95/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a95/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a95 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["K11a95"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$6 t^2-18 t+25-18 t^{-1} +6 t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$6 z^4+6 z^2+1$

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]=

{ 73, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{13}+3 q^{12}-5 q^{11}+7 q^{10}-10 q^9+11 q^8-11 q^7+10 q^6-7 q^5+5 q^4-2 q^3+q^2$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^4 a^{-4} +2 z^4 a^{-6} +2 z^4 a^{-8} +z^4 a^{-10} +2 z^2 a^{-4} +3 z^2 a^{-6} +2 z^2 a^{-8} -z^2 a^{-12} + a^{-4} + a^{-6} - a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10} a^{-10} +z^{10} a^{-12} +2 z^9 a^{-9} +5 z^9 a^{-11} +3 z^9 a^{-13} +3 z^8 a^{-8} +2 z^8 a^{-10} +2 z^8 a^{-12} +3 z^8 a^{-14} +3 z^7 a^{-7} -15 z^7 a^{-11} -11 z^7 a^{-13} +z^7 a^{-15} +3 z^6 a^{-6} -3 z^6 a^{-8} -10 z^6 a^{-10} -17 z^6 a^{-12} -13 z^6 a^{-14} +2 z^5 a^{-5} -z^5 a^{-7} -6 z^5 a^{-9} +10 z^5 a^{-11} +9 z^5 a^{-13} -4 z^5 a^{-15} +z^4 a^{-4} -3 z^4 a^{-6} +2 z^4 a^{-8} +9 z^4 a^{-10} +18 z^4 a^{-12} +15 z^4 a^{-14} -2 z^3 a^{-5} -2 z^3 a^{-7} +5 z^3 a^{-9} +z^3 a^{-11} +4 z^3 a^{-15} -2 z^2 a^{-4} +2 z^2 a^{-6} -z^2 a^{-8} -5 z^2 a^{-10} -4 z^2 a^{-12} -4 z^2 a^{-14} +2 z a^{-7} -z a^{-9} -3 z a^{-11} -z a^{-13} -z a^{-15} + a^{-4} - a^{-6} + a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_53,}

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["K11a95"];

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]=

{ $6 t^2-18 t+25-18 t^{-1} +6 t^{-2}$, $-q^{13}+3 q^{12}-5 q^{11}+7 q^{10}-10 q^9+11 q^8-11 q^7+10 q^6-7 q^5+5 q^4-2 q^3+q^2$ }

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]=

{10_53,}

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:

(6, 15)

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 $24$ $120$ $288$ $732$ $92$ $2880$ $5008$ $800$ $568$ $2304$ $7200$ $17568$ $2208$ $\frac{176751}{5}$ $\frac{25828}{15}$ $\frac{176164}{15}$ $\frac{929}{3}$ $\frac{6991}{5}$

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 $t^rq^j$ 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=$4 is the signature of K11a95. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 10 11 χ 27                       1 -1 25                     2   2 23                   3 1   -2 21                 4 2     2 19               6 3       -3 17             5 4         1 15           6 6           0 13         4 5             -1 11       3 6               3 9     2 4                 -2 7     3                   3 5 1 2                     -1 3 1                       1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=6$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=9$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=10$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=11$ ${\mathbb Z}_2$ ${\mathbb Z}$

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.

K11a94 K11a96