K11a195

K11a194

K11a196

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a195 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a195/ThurstonBennequinNumber Hyperbolic Volume 9.31337 A-Polynomial See Data:K11a195/A-polynomial

[edit Notes for K11a195's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for K11a195's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{2}-13t+21-13t^{-1}+3t^{-2}$
Conway polynomial	$3z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 53, 0 }
Jones polynomial	$-q^{3}+3q^{2}-4q+6-7q^{-1}+8q^{-2}-7q^{-3}+6q^{-4}-5q^{-5}+3q^{-6}-2q^{-7}+q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$a^{8}-2z^{2}a^{6}-2a^{6}+z^{4}a^{4}+z^{2}a^{4}+a^{4}+z^{4}a^{2}+z^{4}+z^{2}+1-z^{2}a^{-2}$
Kauffman polynomial (db, data sources)	$a^{6}z^{10}+a^{4}z^{10}+2a^{7}z^{9}+5a^{5}z^{9}+3a^{3}z^{9}+a^{8}z^{8}-2a^{6}z^{8}+a^{4}z^{8}+4a^{2}z^{8}-12a^{7}z^{7}-26a^{5}z^{7}-10a^{3}z^{7}+4az^{7}-6a^{8}z^{6}-10a^{6}z^{6}-19a^{4}z^{6}-11a^{2}z^{6}+4z^{6}+23a^{7}z^{5}+42a^{5}z^{5}+9a^{3}z^{5}-6az^{5}+4z^{5}a^{-1}+11a^{8}z^{4}+26a^{6}z^{4}+29a^{4}z^{4}+8a^{2}z^{4}+3z^{4}a^{-2}-3z^{4}-17a^{7}z^{3}-27a^{5}z^{3}-6a^{3}z^{3}-3z^{3}a^{-1}+z^{3}a^{-3}-7a^{8}z^{2}-16a^{6}z^{2}-12a^{4}z^{2}-3a^{2}z^{2}-2z^{2}a^{-2}-2z^{2}+4a^{7}z+7a^{5}z+3a^{3}z+a^{8}+2a^{6}+a^{4}+1$
The A2 invariant	Data:K11a195/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a195/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a195 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["K11a195"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{2}-13t+21-13t^{-1}+3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 53, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{3}+3q^{2}-4q+6-7q^{-1}+8q^{-2}-7q^{-3}+6q^{-4}-5q^{-5}+3q^{-6}-2q^{-7}+q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{8}-2z^{2}a^{6}-2a^{6}+z^{4}a^{4}+z^{2}a^{4}+a^{4}+z^{4}a^{2}+z^{4}+z^{2}+1-z^{2}a^{-2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{6}z^{10}+a^{4}z^{10}+2a^{7}z^{9}+5a^{5}z^{9}+3a^{3}z^{9}+a^{8}z^{8}-2a^{6}z^{8}+a^{4}z^{8}+4a^{2}z^{8}-12a^{7}z^{7}-26a^{5}z^{7}-10a^{3}z^{7}+4az^{7}-6a^{8}z^{6}-10a^{6}z^{6}-19a^{4}z^{6}-11a^{2}z^{6}+4z^{6}+23a^{7}z^{5}+42a^{5}z^{5}+9a^{3}z^{5}-6az^{5}+4z^{5}a^{-1}+11a^{8}z^{4}+26a^{6}z^{4}+29a^{4}z^{4}+8a^{2}z^{4}+3z^{4}a^{-2}-3z^{4}-17a^{7}z^{3}-27a^{5}z^{3}-6a^{3}z^{3}-3z^{3}a^{-1}+z^{3}a^{-3}-7a^{8}z^{2}-16a^{6}z^{2}-12a^{4}z^{2}-3a^{2}z^{2}-2z^{2}a^{-2}-2z^{2}+4a^{7}z+7a^{5}z+3a^{3}z+a^{8}+2a^{6}+a^{4}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n114,}

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["K11a195"];

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]=

{ $3t^{2}-13t+21-13t^{-1}+3t^{-2}$, $-q^{3}+3q^{2}-4q+6-7q^{-1}+8q^{-2}-7q^{-3}+6q^{-4}-5q^{-5}+3q^{-6}-2q^{-7}+q^{-8}$ }

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]=

{K11n114,}

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:

(-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 $-4$ $24$ $8$ $-{\frac {254}{3}}$ $-{\frac {58}{3}}$ $-96$ $112$ $-64$ $120$ $-{\frac {32}{3}}$ $288$ ${\frac {1016}{3}}$ ${\frac {232}{3}}$ ${\frac {16049}{30}}$ ${\frac {7222}{15}}$ $-{\frac {5702}{45}}$ $-{\frac {2513}{18}}$ $-{\frac {1231}{30}}$

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 K11a195. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 7                       1 -1 5                     2   2 3                   2 1   -1 1                 4 2     2 -1               4 3       -1 -3             4 3         1 -5           3 4           1 -7         3 4             -1 -9       2 3               1 -11     1 3                 -2 -13   1 2                   1 -15   1                     -1 -17 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a194 K11a196