K11a139

K11a138

K11a140

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a139 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a139/ThurstonBennequinNumber Hyperbolic Volume 13.7305 A-Polynomial See Data:K11a139/A-polynomial

[edit Notes for K11a139's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a139's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-12t^{2}+20t-23+20t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 99, 2 }
Jones polynomial	$-q^{8}+3q^{7}-6q^{6}+10q^{5}-14q^{4}+16q^{3}-15q^{2}+14q-10+6q^{-1}-3q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-6z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-14z^{4}a^{-2}+9z^{4}a^{-4}-z^{4}a^{-6}+4z^{4}-14z^{2}a^{-2}+13z^{2}a^{-4}-3z^{2}a^{-6}+5z^{2}-4a^{-2}+5a^{-4}-2a^{-6}+2$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+6z^{9}a^{-3}+3z^{9}a^{-5}+6z^{8}a^{-2}+7z^{8}a^{-4}+5z^{8}a^{-6}+4z^{8}+3az^{7}-3z^{7}a^{-1}-10z^{7}a^{-3}+z^{7}a^{-5}+5z^{7}a^{-7}+a^{2}z^{6}-20z^{6}a^{-2}-21z^{6}a^{-4}-9z^{6}a^{-6}+3z^{6}a^{-8}-10z^{6}-9az^{5}-5z^{5}a^{-1}+4z^{5}a^{-3}-12z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+25z^{4}a^{-2}+29z^{4}a^{-4}+8z^{4}a^{-6}-6z^{4}a^{-8}+7z^{4}+7az^{3}+5z^{3}a^{-1}+4z^{3}a^{-3}+17z^{3}a^{-5}+9z^{3}a^{-7}-2z^{3}a^{-9}+2a^{2}z^{2}-19z^{2}a^{-2}-19z^{2}a^{-4}-4z^{2}a^{-6}+2z^{2}a^{-8}-4z^{2}-az-2za^{-1}-3za^{-3}-5za^{-5}-3za^{-7}+4a^{-2}+5a^{-4}+2a^{-6}+2$
The A2 invariant	$q^{8}-q^{6}+2q^{4}-q^{2}-1+2q^{-2}-3q^{-4}+4q^{-6}-q^{-8}+q^{-10}+q^{-12}-2q^{-14}+3q^{-16}-q^{-18}-q^{-24}$
The G2 invariant	$q^{46}-2q^{44}+5q^{42}-9q^{40}+10q^{38}-10q^{36}+2q^{34}+15q^{32}-34q^{30}+55q^{28}-63q^{26}+50q^{24}-15q^{22}-44q^{20}+112q^{18}-162q^{16}+172q^{14}-120q^{12}+15q^{10}+115q^{8}-220q^{6}+271q^{4}-231q^{2}+111+47q^{-2}-192q^{-4}+253q^{-6}-214q^{-8}+98q^{-10}+53q^{-12}-158q^{-14}+179q^{-16}-114q^{-18}-25q^{-20}+168q^{-22}-258q^{-24}+232q^{-26}-109q^{-28}-77q^{-30}+265q^{-32}-369q^{-34}+357q^{-36}-226q^{-38}+23q^{-40}+189q^{-42}-339q^{-44}+362q^{-46}-257q^{-48}+90q^{-50}+97q^{-52}-207q^{-54}+215q^{-56}-124q^{-58}-9q^{-60}+126q^{-62}-179q^{-64}+133q^{-66}-18q^{-68}-116q^{-70}+219q^{-72}-235q^{-74}+174q^{-76}-60q^{-78}-77q^{-80}+175q^{-82}-225q^{-84}+203q^{-86}-128q^{-88}+34q^{-90}+55q^{-92}-112q^{-94}+129q^{-96}-112q^{-98}+72q^{-100}-24q^{-102}-18q^{-104}+39q^{-106}-46q^{-108}+39q^{-110}-24q^{-112}+12q^{-114}+q^{-116}-7q^{-118}+7q^{-120}-7q^{-122}+4q^{-124}-2q^{-126}+q^{-128}$

Further Quantum Invariants

K11a139 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["K11a139"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-12t^{2}+20t-23+20t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 99, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+3q^{7}-6q^{6}+10q^{5}-14q^{4}+16q^{3}-15q^{2}+14q-10+6q^{-1}-3q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-6z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-14z^{4}a^{-2}+9z^{4}a^{-4}-z^{4}a^{-6}+4z^{4}-14z^{2}a^{-2}+13z^{2}a^{-4}-3z^{2}a^{-6}+5z^{2}-4a^{-2}+5a^{-4}-2a^{-6}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+6z^{9}a^{-3}+3z^{9}a^{-5}+6z^{8}a^{-2}+7z^{8}a^{-4}+5z^{8}a^{-6}+4z^{8}+3az^{7}-3z^{7}a^{-1}-10z^{7}a^{-3}+z^{7}a^{-5}+5z^{7}a^{-7}+a^{2}z^{6}-20z^{6}a^{-2}-21z^{6}a^{-4}-9z^{6}a^{-6}+3z^{6}a^{-8}-10z^{6}-9az^{5}-5z^{5}a^{-1}+4z^{5}a^{-3}-12z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}-3a^{2}z^{4}+25z^{4}a^{-2}+29z^{4}a^{-4}+8z^{4}a^{-6}-6z^{4}a^{-8}+7z^{4}+7az^{3}+5z^{3}a^{-1}+4z^{3}a^{-3}+17z^{3}a^{-5}+9z^{3}a^{-7}-2z^{3}a^{-9}+2a^{2}z^{2}-19z^{2}a^{-2}-19z^{2}a^{-4}-4z^{2}a^{-6}+2z^{2}a^{-8}-4z^{2}-az-2za^{-1}-3za^{-3}-5za^{-5}-3za^{-7}+4a^{-2}+5a^{-4}+2a^{-6}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a57, K11a108, K11a231,}

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["K11a139"];

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]=

{ $-t^{4}+5t^{3}-12t^{2}+20t-23+20t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$, $-q^{8}+3q^{7}-6q^{6}+10q^{5}-14q^{4}+16q^{3}-15q^{2}+14q-10+6q^{-1}-3q^{-2}+q^{-3}$ }

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]=

{K11a57, K11a108, K11a231,}

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, 3)

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 $4$ $24$ $8$ ${\frac {254}{3}}$ ${\frac {58}{3}}$ $96$ $272$ $-32$ $120$ ${\frac {32}{3}}$ $288$ ${\frac {1016}{3}}$ ${\frac {232}{3}}$ ${\frac {34831}{30}}$ $-{\frac {5062}{15}}$ ${\frac {33062}{45}}$ ${\frac {2225}{18}}$ ${\frac {2671}{30}}$

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 K11a139. 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                     2   2 13                   4 1   -3 11                 6 2     4 9               8 4       -4 7             8 6         2 5           7 8           1 3         7 8             -1 1       4 8               4 -1     2 6                 -4 -3   1 4                   3 -5   2                     -2 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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.

K11a138 K11a140