K11a99

K11a98

K11a100

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a99 at Knotilus!

Knot presentations

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

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:K11a99/ThurstonBennequinNumber Hyperbolic Volume 16.6086 A-Polynomial See Data:K11a99/A-polynomial

[edit Notes for K11a99'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 K11a99's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-17t^{2}+28t-31+28t^{-1}-17t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}-z^{4}-2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 135, 2 }
Jones polynomial	$q^{7}-4q^{6}+9q^{5}-14q^{4}+19q^{3}-22q^{2}+21q-18+14q^{-1}-8q^{-2}+4q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-10z^{4}a^{-2}+3z^{4}a^{-4}+7z^{4}-2a^{2}z^{2}-10z^{2}a^{-2}+3z^{2}a^{-4}+7z^{2}-4a^{-2}+2a^{-4}+3$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+5az^{9}+13z^{9}a^{-1}+8z^{9}a^{-3}+4a^{2}z^{8}+19z^{8}a^{-2}+13z^{8}a^{-4}+10z^{8}+a^{3}z^{7}-12az^{7}-27z^{7}a^{-1}-z^{7}a^{-3}+13z^{7}a^{-5}-14a^{2}z^{6}-62z^{6}a^{-2}-18z^{6}a^{-4}+9z^{6}a^{-6}-49z^{6}-3a^{3}z^{5}+az^{5}-25z^{5}a^{-3}-17z^{5}a^{-5}+4z^{5}a^{-7}+16a^{2}z^{4}+55z^{4}a^{-2}+7z^{4}a^{-4}-8z^{4}a^{-6}+z^{4}a^{-8}+55z^{4}+3a^{3}z^{3}+10az^{3}+15z^{3}a^{-1}+18z^{3}a^{-3}+9z^{3}a^{-5}-z^{3}a^{-7}-6a^{2}z^{2}-23z^{2}a^{-2}-5z^{2}a^{-4}+3z^{2}a^{-6}-21z^{2}-a^{3}z-3az-3za^{-1}-3za^{-3}-2za^{-5}+4a^{-2}+2a^{-4}+3$
The A2 invariant	$-q^{12}+q^{10}+q^{8}-q^{6}+4q^{4}-2q^{2}+2+q^{-2}-4q^{-4}+3q^{-6}-5q^{-8}+3q^{-10}-q^{-14}+3q^{-16}-2q^{-18}+q^{-20}$
The G2 invariant	$q^{60}-3q^{58}+9q^{56}-19q^{54}+28q^{52}-33q^{50}+17q^{48}+26q^{46}-91q^{44}+165q^{42}-204q^{40}+167q^{38}-37q^{36}-174q^{34}+394q^{32}-522q^{30}+480q^{28}-242q^{26}-135q^{24}+516q^{22}-741q^{20}+713q^{18}-405q^{16}-48q^{14}+467q^{12}-683q^{10}+599q^{8}-269q^{6}-156q^{4}+498q^{2}-586+391q^{-2}+18q^{-4}-460q^{-6}+748q^{-8}-756q^{-10}+454q^{-12}+43q^{-14}-573q^{-16}+935q^{-18}-987q^{-20}+703q^{-22}-172q^{-24}-401q^{-26}+798q^{-28}-894q^{-30}+643q^{-32}-188q^{-34}-279q^{-36}+566q^{-38}-560q^{-40}+292q^{-42}+105q^{-44}-431q^{-46}+544q^{-48}-402q^{-50}+68q^{-52}+293q^{-54}-544q^{-56}+602q^{-58}-444q^{-60}+168q^{-62}+140q^{-64}-372q^{-66}+462q^{-68}-420q^{-70}+276q^{-72}-96q^{-74}-65q^{-76}+179q^{-78}-227q^{-80}+213q^{-82}-152q^{-84}+78q^{-86}-5q^{-88}-46q^{-90}+68q^{-92}-74q^{-94}+57q^{-96}-33q^{-98}+13q^{-100}+4q^{-102}-10q^{-104}+11q^{-106}-10q^{-108}+6q^{-110}-3q^{-112}+q^{-114}$

Further Quantum Invariants

K11a99 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["K11a99"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-17t^{2}+28t-31+28t^{-1}-17t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 135, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{7}-4q^{6}+9q^{5}-14q^{4}+19q^{3}-22q^{2}+21q-18+14q^{-1}-8q^{-2}+4q^{-3}-q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}+5az^{9}+13z^{9}a^{-1}+8z^{9}a^{-3}+4a^{2}z^{8}+19z^{8}a^{-2}+13z^{8}a^{-4}+10z^{8}+a^{3}z^{7}-12az^{7}-27z^{7}a^{-1}-z^{7}a^{-3}+13z^{7}a^{-5}-14a^{2}z^{6}-62z^{6}a^{-2}-18z^{6}a^{-4}+9z^{6}a^{-6}-49z^{6}-3a^{3}z^{5}+az^{5}-25z^{5}a^{-3}-17z^{5}a^{-5}+4z^{5}a^{-7}+16a^{2}z^{4}+55z^{4}a^{-2}+7z^{4}a^{-4}-8z^{4}a^{-6}+z^{4}a^{-8}+55z^{4}+3a^{3}z^{3}+10az^{3}+15z^{3}a^{-1}+18z^{3}a^{-3}+9z^{3}a^{-5}-z^{3}a^{-7}-6a^{2}z^{2}-23z^{2}a^{-2}-5z^{2}a^{-4}+3z^{2}a^{-6}-21z^{2}-a^{3}z-3az-3za^{-1}-3za^{-3}-2za^{-5}+4a^{-2}+2a^{-4}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a277,}

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["K11a99"];

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]=

{ $-t^{4}+6t^{3}-17t^{2}+28t-31+28t^{-1}-17t^{-2}+6t^{-3}-t^{-4}$, $q^{7}-4q^{6}+9q^{5}-14q^{4}+19q^{3}-22q^{2}+21q-18+14q^{-1}-8q^{-2}+4q^{-3}-q^{-4}$ }

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]=

{K11a277,}

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:

(-2, -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 $-8$ $-16$ $32$ ${\frac {116}{3}}$ ${\frac {76}{3}}$ $128$ ${\frac {800}{3}}$ ${\frac {224}{3}}$ $80$ $-{\frac {256}{3}}$ $128$ $-{\frac {928}{3}}$ $-{\frac {608}{3}}$ ${\frac {1769}{15}}$ ${\frac {948}{5}}$ $-{\frac {13084}{45}}$ ${\frac {1111}{9}}$ $-{\frac {1591}{15}}$

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 K11a99. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     3   -3 11                   6 1   5 9                 8 3     -5 7               11 6       5 5             11 8         -3 3           10 11           -1 1         9 12             3 -1       5 9               -4 -3     3 9                 6 -5   1 5                   -4 -7   3                     3 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a98 K11a100