K11a19

K11a18

K11a20

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a19 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X12,5,13,6 X2837 X16,9,17,10 X18,12,19,11 X6,13,7,14 X20,16,21,15 X22,17,1,18 X14,20,15,19 X10,22,11,21
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, 5, -11, 6, -3, 7, -10, 8, -5, 9, -6, 10, -8, 11, -9
Dowker-Thistlethwaite code	4 8 12 2 16 18 6 20 22 14 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a19/ThurstonBennequinNumber Hyperbolic Volume 17.7863 A-Polynomial See Data:K11a19/A-polynomial

[edit Notes for K11a19'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 K11a19's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-18t^{2}+33t-39+33t^{-1}-18t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}-2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 155, 2 }
Jones polynomial	$-q^{8}+4q^{7}-9q^{6}+16q^{5}-22q^{4}+25q^{3}-25q^{2}+22q-16+10q^{-1}-4q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-5z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-11z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}+3z^{4}-12z^{2}a^{-2}+9z^{2}a^{-4}-2z^{2}a^{-6}+4z^{2}-5a^{-2}+4a^{-4}-a^{-6}+3$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}a^{-4}+7z^{9}a^{-1}+14z^{9}a^{-3}+7z^{9}a^{-5}+19z^{8}a^{-2}+21z^{8}a^{-4}+10z^{8}a^{-6}+8z^{8}+4az^{7}-6z^{7}a^{-1}-14z^{7}a^{-3}+4z^{7}a^{-5}+8z^{7}a^{-7}+a^{2}z^{6}-55z^{6}a^{-2}-52z^{6}a^{-4}-12z^{6}a^{-6}+4z^{6}a^{-8}-18z^{6}-8az^{5}-11z^{5}a^{-1}-17z^{5}a^{-3}-26z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}-2a^{2}z^{4}+50z^{4}a^{-2}+43z^{4}a^{-4}+5z^{4}a^{-6}-5z^{4}a^{-8}+15z^{4}+5az^{3}+11z^{3}a^{-1}+21z^{3}a^{-3}+23z^{3}a^{-5}+7z^{3}a^{-7}-z^{3}a^{-9}+a^{2}z^{2}-24z^{2}a^{-2}-18z^{2}a^{-4}-z^{2}a^{-6}+2z^{2}a^{-8}-8z^{2}-az-3za^{-1}-6za^{-3}-6za^{-5}-2za^{-7}+5a^{-2}+4a^{-4}+a^{-6}+3$
The A2 invariant	$q^{8}-2q^{6}+4q^{4}-q^{2}+4q^{-2}-6q^{-4}+4q^{-6}-4q^{-8}+q^{-10}+2q^{-12}-3q^{-14}+5q^{-16}-2q^{-18}+q^{-22}-q^{-24}$
The G2 invariant	$q^{46}-3q^{44}+8q^{42}-16q^{40}+23q^{38}-28q^{36}+20q^{34}+11q^{32}-66q^{30}+145q^{28}-216q^{26}+228q^{24}-143q^{22}-63q^{20}+357q^{18}-625q^{16}+753q^{14}-615q^{12}+190q^{10}+409q^{8}-975q^{6}+1270q^{4}-1120q^{2}+547+253q^{-2}-963q^{-4}+1286q^{-6}-1080q^{-8}+449q^{-10}+337q^{-12}-920q^{-14}+1032q^{-16}-632q^{-18}-121q^{-20}+884q^{-22}-1316q^{-24}+1205q^{-26}-568q^{-28}-380q^{-30}+1275q^{-32}-1781q^{-34}+1685q^{-36}-1007q^{-38}-25q^{-40}+1032q^{-42}-1653q^{-44}+1665q^{-46}-1082q^{-48}+174q^{-50}+693q^{-52}-1162q^{-54}+1067q^{-56}-488q^{-58}-276q^{-60}+875q^{-62}-1027q^{-64}+679q^{-66}+6q^{-68}-720q^{-70}+1166q^{-72}-1164q^{-74}+750q^{-76}-106q^{-78}-527q^{-80}+917q^{-82}-982q^{-84}+755q^{-86}-353q^{-88}-53q^{-90}+345q^{-92}-475q^{-94}+442q^{-96}-312q^{-98}+147q^{-100}-94q^{-104}+128q^{-106}-121q^{-108}+88q^{-110}-46q^{-112}+15q^{-114}+8q^{-116}-17q^{-118}+16q^{-120}-13q^{-122}+7q^{-124}-3q^{-126}+q^{-128}$

Further Quantum Invariants

K11a19 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["K11a19"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-18t^{2}+33t-39+33t^{-1}-18t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 155, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+4q^{7}-9q^{6}+16q^{5}-22q^{4}+25q^{3}-25q^{2}+22q-16+10q^{-1}-4q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-5z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-11z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}+3z^{4}-12z^{2}a^{-2}+9z^{2}a^{-4}-2z^{2}a^{-6}+4z^{2}-5a^{-2}+4a^{-4}-a^{-6}+3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}a^{-4}+7z^{9}a^{-1}+14z^{9}a^{-3}+7z^{9}a^{-5}+19z^{8}a^{-2}+21z^{8}a^{-4}+10z^{8}a^{-6}+8z^{8}+4az^{7}-6z^{7}a^{-1}-14z^{7}a^{-3}+4z^{7}a^{-5}+8z^{7}a^{-7}+a^{2}z^{6}-55z^{6}a^{-2}-52z^{6}a^{-4}-12z^{6}a^{-6}+4z^{6}a^{-8}-18z^{6}-8az^{5}-11z^{5}a^{-1}-17z^{5}a^{-3}-26z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}-2a^{2}z^{4}+50z^{4}a^{-2}+43z^{4}a^{-4}+5z^{4}a^{-6}-5z^{4}a^{-8}+15z^{4}+5az^{3}+11z^{3}a^{-1}+21z^{3}a^{-3}+23z^{3}a^{-5}+7z^{3}a^{-7}-z^{3}a^{-9}+a^{2}z^{2}-24z^{2}a^{-2}-18z^{2}a^{-4}-z^{2}a^{-6}+2z^{2}a^{-8}-8z^{2}-az-3za^{-1}-6za^{-3}-6za^{-5}-2za^{-7}+5a^{-2}+4a^{-4}+a^{-6}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a25, K11a281,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a25,}

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["K11a19"];

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]=

{ $-t^{4}+6t^{3}-18t^{2}+33t-39+33t^{-1}-18t^{-2}+6t^{-3}-t^{-4}$, $-q^{8}+4q^{7}-9q^{6}+16q^{5}-22q^{4}+25q^{3}-25q^{2}+22q-16+10q^{-1}-4q^{-2}+q^{-3}$ }

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]=

{K11a25, K11a281,}

In[6]:=

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

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

Out[6]=

{K11a25,}

Vassiliev invariants

V2 and V3:

(-1, 0)

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 $-4$ $0$ $8$ ${\frac {130}{3}}$ ${\frac {62}{3}}$ $0$ $32$ $0$ $0$ $-{\frac {32}{3}}$ $0$ $-{\frac {520}{3}}$ $-{\frac {248}{3}}$ $-{\frac {7471}{30}}$ ${\frac {1822}{15}}$ $-{\frac {17822}{45}}$ ${\frac {1135}{18}}$ $-{\frac {2671}{30}}$

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 K11a19. 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                 10 3     7 9               12 6       -6 7             13 10         3 5           12 12           0 3         10 13             -3 1       7 13               6 -1     3 9                 -6 -3   1 7                   6 -5   3                     -3 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a18 K11a20