K11a221

K11a220

K11a222

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a221 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a221/ThurstonBennequinNumber Hyperbolic Volume 12.2428 A-Polynomial See Data:K11a221/A-polynomial

[edit Notes for K11a221's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a221's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-10t^{2}+15t-17+15t^{-1}-10t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 79, -2 }
Jones polynomial	$q^{3}-3q^{2}+5q-8+11q^{-1}-12q^{-2}+13q^{-3}-10q^{-4}+8q^{-5}-5q^{-6}+2q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+2a^{4}z^{6}-6a^{2}z^{6}+z^{6}-a^{6}z^{4}+10a^{4}z^{4}-13a^{2}z^{4}+4z^{4}-4a^{6}z^{2}+16a^{4}z^{2}-12a^{2}z^{2}+4z^{2}-4a^{6}+8a^{4}-4a^{2}+1$
Kauffman polynomial (db, data sources)	$a^{4}z^{10}+a^{2}z^{10}+2a^{5}z^{9}+5a^{3}z^{9}+3az^{9}+3a^{6}z^{8}+2a^{4}z^{8}+3a^{2}z^{8}+4z^{8}+3a^{7}z^{7}-a^{5}z^{7}-14a^{3}z^{7}-7az^{7}+3z^{7}a^{-1}+2a^{8}z^{6}-6a^{6}z^{6}-11a^{4}z^{6}-17a^{2}z^{6}+z^{6}a^{-2}-13z^{6}+a^{9}z^{5}-6a^{7}z^{5}-4a^{5}z^{5}+14a^{3}z^{5}+az^{5}-10z^{5}a^{-1}-4a^{8}z^{4}+9a^{6}z^{4}+24a^{4}z^{4}+25a^{2}z^{4}-3z^{4}a^{-2}+11z^{4}-3a^{9}z^{3}+4a^{7}z^{3}+11a^{5}z^{3}+2a^{3}z^{3}+5az^{3}+7z^{3}a^{-1}+a^{8}z^{2}-10a^{6}z^{2}-21a^{4}z^{2}-15a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+2a^{9}z-2a^{7}z-7a^{5}z-5a^{3}z-3az-za^{-1}+4a^{6}+8a^{4}+4a^{2}+1$
The A2 invariant	$-q^{24}-q^{22}-q^{20}-2q^{18}+2q^{16}+3q^{12}+3q^{10}+3q^{6}-3q^{4}+q^{2}-1-q^{-2}+q^{-4}-q^{-6}+q^{-8}$
The G2 invariant	$q^{128}-q^{126}+3q^{124}-4q^{122}+4q^{120}-3q^{118}-q^{116}+8q^{114}-15q^{112}+21q^{110}-22q^{108}+14q^{106}-2q^{104}-18q^{102}+39q^{100}-53q^{98}+53q^{96}-41q^{94}+9q^{92}+23q^{90}-57q^{88}+78q^{86}-84q^{84}+66q^{82}-32q^{80}-20q^{78}+63q^{76}-91q^{74}+91q^{72}-62q^{70}+11q^{68}+39q^{66}-70q^{64}+68q^{62}-27q^{60}-25q^{58}+76q^{56}-83q^{54}+50q^{52}+20q^{50}-89q^{48}+143q^{46}-137q^{44}+89q^{42}-q^{40}-84q^{38}+151q^{36}-164q^{34}+128q^{32}-53q^{30}-32q^{28}+98q^{26}-124q^{24}+104q^{22}-46q^{20}-23q^{18}+69q^{16}-85q^{14}+50q^{12}+10q^{10}-72q^{8}+103q^{6}-93q^{4}+38q^{2}+35-100q^{-2}+128q^{-4}-112q^{-6}+63q^{-8}+2q^{-10}-61q^{-12}+93q^{-14}-91q^{-16}+68q^{-18}-28q^{-20}-8q^{-22}+31q^{-24}-40q^{-26}+36q^{-28}-24q^{-30}+12q^{-32}+q^{-34}-7q^{-36}+7q^{-38}-7q^{-40}+4q^{-42}-2q^{-44}+q^{-46}$

Further Quantum Invariants

K11a221 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["K11a221"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-10t^{2}+15t-17+15t^{-1}-10t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-3z^{6}+4z^{2}+1$

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]=

{ 79, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{3}-3q^{2}+5q-8+11q^{-1}-12q^{-2}+13q^{-3}-10q^{-4}+8q^{-5}-5q^{-6}+2q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+2a^{4}z^{6}-6a^{2}z^{6}+z^{6}-a^{6}z^{4}+10a^{4}z^{4}-13a^{2}z^{4}+4z^{4}-4a^{6}z^{2}+16a^{4}z^{2}-12a^{2}z^{2}+4z^{2}-4a^{6}+8a^{4}-4a^{2}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{4}z^{10}+a^{2}z^{10}+2a^{5}z^{9}+5a^{3}z^{9}+3az^{9}+3a^{6}z^{8}+2a^{4}z^{8}+3a^{2}z^{8}+4z^{8}+3a^{7}z^{7}-a^{5}z^{7}-14a^{3}z^{7}-7az^{7}+3z^{7}a^{-1}+2a^{8}z^{6}-6a^{6}z^{6}-11a^{4}z^{6}-17a^{2}z^{6}+z^{6}a^{-2}-13z^{6}+a^{9}z^{5}-6a^{7}z^{5}-4a^{5}z^{5}+14a^{3}z^{5}+az^{5}-10z^{5}a^{-1}-4a^{8}z^{4}+9a^{6}z^{4}+24a^{4}z^{4}+25a^{2}z^{4}-3z^{4}a^{-2}+11z^{4}-3a^{9}z^{3}+4a^{7}z^{3}+11a^{5}z^{3}+2a^{3}z^{3}+5az^{3}+7z^{3}a^{-1}+a^{8}z^{2}-10a^{6}z^{2}-21a^{4}z^{2}-15a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+2a^{9}z-2a^{7}z-7a^{5}z-5a^{3}z-3az-za^{-1}+4a^{6}+8a^{4}+4a^{2}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a259,}

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["K11a221"];

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]=

{ $-t^{4}+5t^{3}-10t^{2}+15t-17+15t^{-1}-10t^{-2}+5t^{-3}-t^{-4}$, $q^{3}-3q^{2}+5q-8+11q^{-1}-12q^{-2}+13q^{-3}-10q^{-4}+8q^{-5}-5q^{-6}+2q^{-7}-q^{-8}$ }

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]=

{K11a259,}

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:

(4, -8)

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 $16$ $-64$ $128$ ${\frac {968}{3}}$ ${\frac {184}{3}}$ $-1024$ $-{\frac {5536}{3}}$ $-{\frac {736}{3}}$ $-384$ ${\frac {2048}{3}}$ $2048$ ${\frac {15488}{3}}$ ${\frac {2944}{3}}$ ${\frac {157382}{15}}$ $-{\frac {8648}{15}}$ ${\frac {229688}{45}}$ ${\frac {1834}{9}}$ ${\frac {11222}{15}}$

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 K11a221. 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                     2   -2 3                   3 1   2 1                 5 2     -3 -1               6 3       3 -3             7 6         -1 -5           6 5           1 -7         4 7             3 -9       4 6               -2 -11     1 4                 3 -13   1 4                   -3 -15   1                     1 -17 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

K11a220 K11a222