K11a216

K11a215

K11a217

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a216 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a216/ThurstonBennequinNumber Hyperbolic Volume 17.2795 A-Polynomial See Data:K11a216/A-polynomial

[edit Notes for K11a216's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $1$ Rasmussen s-Invariant -2

[edit Notes for K11a216's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-17t^{2}+31t-37+31t^{-1}-17t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}-z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 147, 2 }
Jones polynomial	$-q^{8}+4q^{7}-9q^{6}+15q^{5}-21q^{4}+24q^{3}-23q^{2}+21q-15+9q^{-1}-4q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-5z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-10z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}+3z^{4}-8z^{2}a^{-2}+8z^{2}a^{-4}-2z^{2}a^{-6}+3z^{2}-a^{-2}+2a^{-4}-a^{-6}+1$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}a^{-4}+6z^{9}a^{-1}+13z^{9}a^{-3}+7z^{9}a^{-5}+15z^{8}a^{-2}+18z^{8}a^{-4}+10z^{8}a^{-6}+7z^{8}+4az^{7}-4z^{7}a^{-1}-16z^{7}a^{-3}+8z^{7}a^{-7}+a^{2}z^{6}-44z^{6}a^{-2}-47z^{6}a^{-4}-15z^{6}a^{-6}+4z^{6}a^{-8}-15z^{6}-9az^{5}-13z^{5}a^{-1}-9z^{5}a^{-3}-18z^{5}a^{-5}-12z^{5}a^{-7}+z^{5}a^{-9}-2a^{2}z^{4}+38z^{4}a^{-2}+41z^{4}a^{-4}+10z^{4}a^{-6}-5z^{4}a^{-8}+10z^{4}+6az^{3}+12z^{3}a^{-1}+16z^{3}a^{-3}+18z^{3}a^{-5}+7z^{3}a^{-7}-z^{3}a^{-9}+a^{2}z^{2}-15z^{2}a^{-2}-15z^{2}a^{-4}-4z^{2}a^{-6}+z^{2}a^{-8}-4z^{2}-az-3za^{-1}-4za^{-3}-4za^{-5}-2za^{-7}+a^{-2}+2a^{-4}+a^{-6}+1$
The A2 invariant	$q^{8}-2q^{6}+3q^{4}-2q^{2}-1+4q^{-2}-4q^{-4}+6q^{-6}-2q^{-8}+q^{-10}+q^{-12}-4q^{-14}+4q^{-16}-2q^{-18}+q^{-22}-q^{-24}$
The G2 invariant	$q^{46}-3q^{44}+8q^{42}-16q^{40}+22q^{38}-25q^{36}+14q^{34}+18q^{32}-66q^{30}+129q^{28}-176q^{26}+173q^{24}-98q^{22}-67q^{20}+290q^{18}-494q^{16}+587q^{14}-481q^{12}+158q^{10}+304q^{8}-746q^{6}+998q^{4}-914q^{2}+492+127q^{-2}-723q^{-4}+1027q^{-6}-919q^{-8}+449q^{-10}+188q^{-12}-688q^{-14}+841q^{-16}-570q^{-18}-12q^{-20}+643q^{-22}-1038q^{-24}+998q^{-26}-517q^{-28}-239q^{-30}+989q^{-32}-1425q^{-34}+1393q^{-36}-879q^{-38}+60q^{-40}+769q^{-42}-1323q^{-44}+1389q^{-46}-970q^{-48}+253q^{-50}+481q^{-52}-920q^{-54}+915q^{-56}-499q^{-58}-114q^{-60}+636q^{-62}-836q^{-64}+614q^{-66}-87q^{-68}-512q^{-70}+930q^{-72}-980q^{-74}+679q^{-76}-154q^{-78}-393q^{-80}+751q^{-82}-847q^{-84}+677q^{-86}-341q^{-88}-17q^{-90}+288q^{-92}-413q^{-94}+398q^{-96}-287q^{-98}+143q^{-100}-9q^{-102}-82q^{-104}+116q^{-106}-114q^{-108}+83q^{-110}-45q^{-112}+16q^{-114}+7q^{-116}-16q^{-118}+16q^{-120}-13q^{-122}+7q^{-124}-3q^{-126}+q^{-128}$

Further Quantum Invariants

K11a216 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["K11a216"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-17t^{2}+31t-37+31t^{-1}-17t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 147, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+4q^{7}-9q^{6}+15q^{5}-21q^{4}+24q^{3}-23q^{2}+21q-15+9q^{-1}-4q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-5z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-10z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}+3z^{4}-8z^{2}a^{-2}+8z^{2}a^{-4}-2z^{2}a^{-6}+3z^{2}-a^{-2}+2a^{-4}-a^{-6}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}a^{-4}+6z^{9}a^{-1}+13z^{9}a^{-3}+7z^{9}a^{-5}+15z^{8}a^{-2}+18z^{8}a^{-4}+10z^{8}a^{-6}+7z^{8}+4az^{7}-4z^{7}a^{-1}-16z^{7}a^{-3}+8z^{7}a^{-7}+a^{2}z^{6}-44z^{6}a^{-2}-47z^{6}a^{-4}-15z^{6}a^{-6}+4z^{6}a^{-8}-15z^{6}-9az^{5}-13z^{5}a^{-1}-9z^{5}a^{-3}-18z^{5}a^{-5}-12z^{5}a^{-7}+z^{5}a^{-9}-2a^{2}z^{4}+38z^{4}a^{-2}+41z^{4}a^{-4}+10z^{4}a^{-6}-5z^{4}a^{-8}+10z^{4}+6az^{3}+12z^{3}a^{-1}+16z^{3}a^{-3}+18z^{3}a^{-5}+7z^{3}a^{-7}-z^{3}a^{-9}+a^{2}z^{2}-15z^{2}a^{-2}-15z^{2}a^{-4}-4z^{2}a^{-6}+z^{2}a^{-8}-4z^{2}-az-3za^{-1}-4za^{-3}-4za^{-5}-2za^{-7}+a^{-2}+2a^{-4}+a^{-6}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a196, K11a286,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a196,}

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["K11a216"];

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]=

{ $-t^{4}+6t^{3}-17t^{2}+31t-37+31t^{-1}-17t^{-2}+6t^{-3}-t^{-4}$, $-q^{8}+4q^{7}-9q^{6}+15q^{5}-21q^{4}+24q^{3}-23q^{2}+21q-15+9q^{-1}-4q^{-2}+q^{-3}$ }

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]=

{K11a196, K11a286,}

In[6]:=

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

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

Out[6]=

{K11a196,}

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 $4$ $16$ $8$ ${\frac {158}{3}}$ ${\frac {34}{3}}$ $64$ ${\frac {448}{3}}$ $-{\frac {32}{3}}$ $48$ ${\frac {32}{3}}$ $128$ ${\frac {632}{3}}$ ${\frac {136}{3}}$ ${\frac {16591}{30}}$ $-{\frac {234}{5}}$ ${\frac {10142}{45}}$ ${\frac {689}{18}}$ ${\frac {271}{30}}$

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 K11a216. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     3   3 13                   6 1   -5 11                 9 3     6 9               12 6       -6 7             12 9         3 5           11 12           1 3         10 12             -2 1       6 12               6 -1     3 9                 -6 -3   1 6                   5 -5   3                     -3 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a215 K11a217