K11a121

K11a120

K11a122

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a121 at Knotilus!

Knot presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a121/ThurstonBennequinNumber Hyperbolic Volume 14.7555 A-Polynomial See Data:K11a121/A-polynomial

[edit Notes for K11a121's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for K11a121's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-9t^{2}+29t-41+29t^{-1}-9t^{-2}+t^{-3}$
Conway polynomial	$z^{6}-3z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 119, -2 }
Jones polynomial	$q^{3}-4q^{2}+8q-12+17q^{-1}-19q^{-2}+19q^{-3}-16q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{8}+3z^{2}a^{6}+2a^{6}-3z^{4}a^{4}-4z^{2}a^{4}-2a^{4}+z^{6}a^{2}+2z^{4}a^{2}+4z^{2}a^{2}+2a^{2}-2z^{4}-2z^{2}+z^{2}a^{-2}$
Kauffman polynomial (db, data sources)	$a^{4}z^{10}+a^{2}z^{10}+3a^{5}z^{9}+7a^{3}z^{9}+4az^{9}+5a^{6}z^{8}+10a^{4}z^{8}+11a^{2}z^{8}+6z^{8}+5a^{7}z^{7}+7a^{5}z^{7}-3a^{3}z^{7}-az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-2a^{6}z^{6}-15a^{4}z^{6}-25a^{2}z^{6}+z^{6}a^{-2}-14z^{6}+a^{9}z^{5}-7a^{7}z^{5}-19a^{5}z^{5}-13a^{3}z^{5}-12az^{5}-10z^{5}a^{-1}-5a^{8}z^{4}-6a^{6}z^{4}+11a^{2}z^{4}-2z^{4}a^{-2}+8z^{4}-2a^{9}z^{3}+4a^{7}z^{3}+14a^{5}z^{3}+9a^{3}z^{3}+7az^{3}+6z^{3}a^{-1}+3a^{8}z^{2}+7a^{6}z^{2}+7a^{4}z^{2}+2a^{2}z^{2}+z^{2}a^{-2}+a^{9}z-a^{7}z-3a^{5}z-a^{3}z-a^{8}-2a^{6}-2a^{4}-2a^{2}$
The A2 invariant	Data:K11a121/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a121/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a121 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["K11a121"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-9t^{2}+29t-41+29t^{-1}-9t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}-3z^{4}+2z^{2}+1$

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]=

{ 119, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-4q^{2}+8q-12+17q^{-1}-19q^{-2}+19q^{-3}-16q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{4}z^{10}+a^{2}z^{10}+3a^{5}z^{9}+7a^{3}z^{9}+4az^{9}+5a^{6}z^{8}+10a^{4}z^{8}+11a^{2}z^{8}+6z^{8}+5a^{7}z^{7}+7a^{5}z^{7}-3a^{3}z^{7}-az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-2a^{6}z^{6}-15a^{4}z^{6}-25a^{2}z^{6}+z^{6}a^{-2}-14z^{6}+a^{9}z^{5}-7a^{7}z^{5}-19a^{5}z^{5}-13a^{3}z^{5}-12az^{5}-10z^{5}a^{-1}-5a^{8}z^{4}-6a^{6}z^{4}+11a^{2}z^{4}-2z^{4}a^{-2}+8z^{4}-2a^{9}z^{3}+4a^{7}z^{3}+14a^{5}z^{3}+9a^{3}z^{3}+7az^{3}+6z^{3}a^{-1}+3a^{8}z^{2}+7a^{6}z^{2}+7a^{4}z^{2}+2a^{2}z^{2}+z^{2}a^{-2}+a^{9}z-a^{7}z-3a^{5}z-a^{3}z-a^{8}-2a^{6}-2a^{4}-2a^{2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a66,}

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["K11a121"];

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]=

{ $t^{3}-9t^{2}+29t-41+29t^{-1}-9t^{-2}+t^{-3}$, $q^{3}-4q^{2}+8q-12+17q^{-1}-19q^{-2}+19q^{-3}-16q^{-4}+12q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$ }

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]=

{}

In[6]:=

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

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

Out[6]=

{K11a66,}

Vassiliev invariants

V2 and V3:

(2, -4)

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 $8$ $-32$ $32$ ${\frac {460}{3}}$ ${\frac {116}{3}}$ $-256$ $-{\frac {2144}{3}}$ $-{\frac {224}{3}}$ $-224$ ${\frac {256}{3}}$ $512$ ${\frac {3680}{3}}$ ${\frac {928}{3}}$ ${\frac {49111}{15}}$ $-{\frac {11884}{15}}$ ${\frac {105724}{45}}$ ${\frac {1145}{9}}$ ${\frac {6151}{15}}$

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 K11a121. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a120 K11a122