K11a125

K11a124

K11a126

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a125 at Knotilus!

Knot presentations

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

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:K11a125/ThurstonBennequinNumber Hyperbolic Volume 18.6455 A-Polynomial See Data:K11a125/A-polynomial

[edit Notes for K11a125'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 K11a125's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-19t^{2}+38t-47+38t^{-1}-19t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}-3z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 175, 2 }
Jones polynomial	$-q^{8}+5q^{7}-12q^{6}+19q^{5}-25q^{4}+29q^{3}-28q^{2}+24q-17+10q^{-1}-4q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-5z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-11z^{4}a^{-2}+6z^{4}a^{-4}-z^{4}a^{-6}+3z^{4}-10z^{2}a^{-2}+7z^{2}a^{-4}-z^{2}a^{-6}+4z^{2}-3a^{-2}+3a^{-4}-a^{-6}+2$
Kauffman polynomial (db, data sources)	$3z^{10}a^{-2}+3z^{10}a^{-4}+8z^{9}a^{-1}+19z^{9}a^{-3}+11z^{9}a^{-5}+18z^{8}a^{-2}+26z^{8}a^{-4}+16z^{8}a^{-6}+8z^{8}+4az^{7}-9z^{7}a^{-1}-27z^{7}a^{-3}-2z^{7}a^{-5}+12z^{7}a^{-7}+a^{2}z^{6}-55z^{6}a^{-2}-66z^{6}a^{-4}-24z^{6}a^{-6}+5z^{6}a^{-8}-17z^{6}-8az^{5}-5z^{5}a^{-1}-3z^{5}a^{-3}-22z^{5}a^{-5}-15z^{5}a^{-7}+z^{5}a^{-9}-2a^{2}z^{4}+50z^{4}a^{-2}+49z^{4}a^{-4}+12z^{4}a^{-6}-3z^{4}a^{-8}+14z^{4}+5az^{3}+8z^{3}a^{-1}+12z^{3}a^{-3}+14z^{3}a^{-5}+5z^{3}a^{-7}+a^{2}z^{2}-20z^{2}a^{-2}-16z^{2}a^{-4}-4z^{2}a^{-6}-7z^{2}-az-2za^{-1}-2za^{-3}-2za^{-5}-za^{-7}+3a^{-2}+3a^{-4}+a^{-6}+2$
The A2 invariant	$q^{8}-2q^{6}+4q^{4}-2q^{2}-1+5q^{-2}-6q^{-4}+5q^{-6}-3q^{-8}+q^{-10}+3q^{-12}-4q^{-14}+5q^{-16}-3q^{-18}-q^{-20}+2q^{-22}-q^{-24}$
The G2 invariant	$q^{46}-3q^{44}+8q^{42}-16q^{40}+23q^{38}-28q^{36}+20q^{34}+10q^{32}-62q^{30}+137q^{28}-208q^{26}+233q^{24}-176q^{22}-2q^{20}+294q^{18}-608q^{16}+833q^{14}-812q^{12}+451q^{10}+199q^{8}-958q^{6}+1532q^{4}-1632q^{2}+1151-188q^{-2}-916q^{-4}+1717q^{-6}-1870q^{-8}+1282q^{-10}-198q^{-12}-912q^{-14}+1540q^{-16}-1415q^{-18}+595q^{-20}+565q^{-22}-1513q^{-24}+1825q^{-26}-1319q^{-28}+142q^{-30}+1226q^{-32}-2263q^{-34}+2534q^{-36}-1901q^{-38}+590q^{-40}+955q^{-42}-2166q^{-44}+2616q^{-46}-2157q^{-48}+971q^{-50}+450q^{-52}-1558q^{-54}+1919q^{-56}-1425q^{-58}+370q^{-60}+777q^{-62}-1465q^{-64}+1405q^{-66}-655q^{-68}-462q^{-70}+1422q^{-72}-1815q^{-74}+1496q^{-76}-606q^{-78}-462q^{-80}+1302q^{-82}-1636q^{-84}+1413q^{-86}-801q^{-88}+61q^{-90}+535q^{-92}-847q^{-94}+837q^{-96}-597q^{-98}+284q^{-100}+7q^{-102}-189q^{-104}+249q^{-106}-230q^{-108}+153q^{-110}-72q^{-112}+14q^{-114}+22q^{-116}-31q^{-118}+28q^{-120}-20q^{-122}+10q^{-124}-4q^{-126}+q^{-128}$

Further Quantum Invariants

K11a125 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["K11a125"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-19t^{2}+38t-47+38t^{-1}-19t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 175, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+5q^{7}-12q^{6}+19q^{5}-25q^{4}+29q^{3}-28q^{2}+24q-17+10q^{-1}-4q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-5z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-11z^{4}a^{-2}+6z^{4}a^{-4}-z^{4}a^{-6}+3z^{4}-10z^{2}a^{-2}+7z^{2}a^{-4}-z^{2}a^{-6}+4z^{2}-3a^{-2}+3a^{-4}-a^{-6}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3z^{10}a^{-2}+3z^{10}a^{-4}+8z^{9}a^{-1}+19z^{9}a^{-3}+11z^{9}a^{-5}+18z^{8}a^{-2}+26z^{8}a^{-4}+16z^{8}a^{-6}+8z^{8}+4az^{7}-9z^{7}a^{-1}-27z^{7}a^{-3}-2z^{7}a^{-5}+12z^{7}a^{-7}+a^{2}z^{6}-55z^{6}a^{-2}-66z^{6}a^{-4}-24z^{6}a^{-6}+5z^{6}a^{-8}-17z^{6}-8az^{5}-5z^{5}a^{-1}-3z^{5}a^{-3}-22z^{5}a^{-5}-15z^{5}a^{-7}+z^{5}a^{-9}-2a^{2}z^{4}+50z^{4}a^{-2}+49z^{4}a^{-4}+12z^{4}a^{-6}-3z^{4}a^{-8}+14z^{4}+5az^{3}+8z^{3}a^{-1}+12z^{3}a^{-3}+14z^{3}a^{-5}+5z^{3}a^{-7}+a^{2}z^{2}-20z^{2}a^{-2}-16z^{2}a^{-4}-4z^{2}a^{-6}-7z^{2}-az-2za^{-1}-2za^{-3}-2za^{-5}-za^{-7}+3a^{-2}+3a^{-4}+a^{-6}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a297,}

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["K11a125"];

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]=

{ $-t^{4}+6t^{3}-19t^{2}+38t-47+38t^{-1}-19t^{-2}+6t^{-3}-t^{-4}$, $-q^{8}+5q^{7}-12q^{6}+19q^{5}-25q^{4}+29q^{3}-28q^{2}+24q-17+10q^{-1}-4q^{-2}+q^{-3}$ }

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]=

{}

In[6]:=

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

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

Out[6]=

{K11a297,}

Vassiliev invariants

V2 and V3:

(0, 1)

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 $0$ $8$ $0$ $48$ $24$ $0$ ${\frac {272}{3}}$ ${\frac {32}{3}}$ $40$ $0$ $32$ $0$ $0$ $168$ $72$ $-24$ $40$ $-40$

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 K11a125. 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                     4   4 13                   8 1   -7 11                 11 4     7 9               14 8       -6 7             15 11         4 5           13 14           1 3         11 15             -4 1       7 14               7 -1     3 10                 -7 -3   1 7                   6 -5   3                     -3 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{15}$ $r=3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a124 K11a126