K11a279

K11a278

K11a280

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a279 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,3,11,4 X16,5,17,6 X12,8,13,7 X18,9,19,10 X2,11,3,12 X20,14,21,13 X22,16,1,15 X4,17,5,18 X8,19,9,20 X14,22,15,21
Gauss code	1, -6, 2, -9, 3, -1, 4, -10, 5, -2, 6, -4, 7, -11, 8, -3, 9, -5, 10, -7, 11, -8
Dowker-Thistlethwaite code	6 10 16 12 18 2 20 22 4 8 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:K11a279/ThurstonBennequinNumber Hyperbolic Volume 14.416 A-Polynomial See Data:K11a279/A-polynomial

[edit Notes for K11a279'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 K11a279's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{3}-12t^{2}+20t-21+20t^{-1}-12t^{-2}+3t^{-3}$
Conway polynomial	$3z^{6}+6z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 91, 2 }
Jones polynomial	$-q^{6}+3q^{5}-6q^{4}+10q^{3}-12q^{2}+14q-14+12q^{-1}-9q^{-2}+6q^{-3}-3q^{-4}+q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+2z^{6}-3a^{2}z^{4}+2z^{4}a^{-2}-z^{4}a^{-4}+8z^{4}+a^{4}z^{2}-9a^{2}z^{2}-z^{2}a^{-2}-2z^{2}a^{-4}+10z^{2}+2a^{4}-5a^{2}-a^{-2}+5$
Kauffman polynomial (db, data sources)	$2a^{2}z^{10}+2z^{10}+3a^{3}z^{9}+10az^{9}+7z^{9}a^{-1}+a^{4}z^{8}-3a^{2}z^{8}+11z^{8}a^{-2}+7z^{8}-15a^{3}z^{7}-42az^{7}-15z^{7}a^{-1}+12z^{7}a^{-3}-5a^{4}z^{6}-18a^{2}z^{6}-27z^{6}a^{-2}+10z^{6}a^{-4}-50z^{6}+24a^{3}z^{5}+51az^{5}-4z^{5}a^{-1}-25z^{5}a^{-3}+6z^{5}a^{-5}+9a^{4}z^{4}+44a^{2}z^{4}+14z^{4}a^{-2}-15z^{4}a^{-4}+3z^{4}a^{-6}+67z^{4}-13a^{3}z^{3}-18az^{3}+11z^{3}a^{-1}+12z^{3}a^{-3}-3z^{3}a^{-5}+z^{3}a^{-7}-7a^{4}z^{2}-28a^{2}z^{2}-4z^{2}a^{-2}+6z^{2}a^{-4}-31z^{2}+2a^{3}z+az-3za^{-1}-2za^{-3}+2a^{4}+5a^{2}+a^{-2}+5$
The A2 invariant	Data:K11a279/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a279/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a279 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["K11a279"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{3}-12t^{2}+20t-21+20t^{-1}-12t^{-2}+3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 91, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{6}+3q^{5}-6q^{4}+10q^{3}-12q^{2}+14q-14+12q^{-1}-9q^{-2}+6q^{-3}-3q^{-4}+q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+2z^{6}-3a^{2}z^{4}+2z^{4}a^{-2}-z^{4}a^{-4}+8z^{4}+a^{4}z^{2}-9a^{2}z^{2}-z^{2}a^{-2}-2z^{2}a^{-4}+10z^{2}+2a^{4}-5a^{2}-a^{-2}+5$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{2}z^{10}+2z^{10}+3a^{3}z^{9}+10az^{9}+7z^{9}a^{-1}+a^{4}z^{8}-3a^{2}z^{8}+11z^{8}a^{-2}+7z^{8}-15a^{3}z^{7}-42az^{7}-15z^{7}a^{-1}+12z^{7}a^{-3}-5a^{4}z^{6}-18a^{2}z^{6}-27z^{6}a^{-2}+10z^{6}a^{-4}-50z^{6}+24a^{3}z^{5}+51az^{5}-4z^{5}a^{-1}-25z^{5}a^{-3}+6z^{5}a^{-5}+9a^{4}z^{4}+44a^{2}z^{4}+14z^{4}a^{-2}-15z^{4}a^{-4}+3z^{4}a^{-6}+67z^{4}-13a^{3}z^{3}-18az^{3}+11z^{3}a^{-1}+12z^{3}a^{-3}-3z^{3}a^{-5}+z^{3}a^{-7}-7a^{4}z^{2}-28a^{2}z^{2}-4z^{2}a^{-2}+6z^{2}a^{-4}-31z^{2}+2a^{3}z+az-3za^{-1}-2za^{-3}+2a^{4}+5a^{2}+a^{-2}+5$

"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["K11a279"];

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]=

{ $3t^{3}-12t^{2}+20t-21+20t^{-1}-12t^{-2}+3t^{-3}$, $-q^{6}+3q^{5}-6q^{4}+10q^{3}-12q^{2}+14q-14+12q^{-1}-9q^{-2}+6q^{-3}-3q^{-4}+q^{-5}$ }

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:

(-1, 2)

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 $-4$ $16$ $8$ $-{\frac {158}{3}}$ $-{\frac {130}{3}}$ $-64$ $-{\frac {224}{3}}$ $-{\frac {416}{3}}$ $112$ $-{\frac {32}{3}}$ $128$ ${\frac {632}{3}}$ ${\frac {520}{3}}$ ${\frac {14129}{30}}$ $-{\frac {2818}{15}}$ ${\frac {30658}{45}}$ $-{\frac {3377}{18}}$ ${\frac {3569}{30}}$

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=$2 is the signature of K11a279. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a278 K11a280