K11a127

K11a126

K11a128

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a127 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 2 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a127/ThurstonBennequinNumber Hyperbolic Volume 16.6559 A-Polynomial See Data:K11a127/A-polynomial

[edit Notes for K11a127's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a127's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+17t^{2}-28t+33-28t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}+z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 137, 4 }
Jones polynomial	$q^{10}-4q^{9}+9q^{8}-15q^{7}+19q^{6}-22q^{5}+22q^{4}-18q^{3}+14q^{2}-8q+4-q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-z^{6}a^{-2}+5z^{6}a^{-4}-2z^{6}a^{-6}-3z^{4}a^{-2}+10z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}-2z^{2}a^{-2}+10z^{2}a^{-4}-8z^{2}a^{-6}+2z^{2}a^{-8}+4a^{-4}-4a^{-6}+a^{-8}$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-4}+2z^{10}a^{-6}+5z^{9}a^{-3}+13z^{9}a^{-5}+8z^{9}a^{-7}+4z^{8}a^{-2}+10z^{8}a^{-4}+20z^{8}a^{-6}+14z^{8}a^{-8}+z^{7}a^{-1}-12z^{7}a^{-3}-25z^{7}a^{-5}+2z^{7}a^{-7}+14z^{7}a^{-9}-14z^{6}a^{-2}-48z^{6}a^{-4}-64z^{6}a^{-6}-21z^{6}a^{-8}+9z^{6}a^{-10}-3z^{5}a^{-1}+z^{5}a^{-3}-8z^{5}a^{-5}-36z^{5}a^{-7}-20z^{5}a^{-9}+4z^{5}a^{-11}+16z^{4}a^{-2}+53z^{4}a^{-4}+54z^{4}a^{-6}+9z^{4}a^{-8}-7z^{4}a^{-10}+z^{4}a^{-12}+3z^{3}a^{-1}+11z^{3}a^{-3}+27z^{3}a^{-5}+32z^{3}a^{-7}+12z^{3}a^{-9}-z^{3}a^{-11}-6z^{2}a^{-2}-21z^{2}a^{-4}-20z^{2}a^{-6}-3z^{2}a^{-8}+2z^{2}a^{-10}-za^{-1}-4za^{-3}-10za^{-5}-10za^{-7}-3za^{-9}+4a^{-4}+4a^{-6}+a^{-8}$
The A2 invariant	$-q^{2}+2-2q^{-2}+2q^{-4}+2q^{-6}-2q^{-8}+6q^{-10}-3q^{-12}+3q^{-14}-q^{-16}-3q^{-18}+2q^{-20}-4q^{-22}+2q^{-24}-q^{-28}+q^{-30}$
The G2 invariant	$q^{12}-3q^{10}+9q^{8}-19q^{6}+28q^{4}-33q^{2}+17+26q^{-2}-91q^{-4}+164q^{-6}-204q^{-8}+168q^{-10}-41q^{-12}-167q^{-14}+393q^{-16}-528q^{-18}+497q^{-20}-263q^{-22}-119q^{-24}+510q^{-26}-760q^{-28}+752q^{-30}-457q^{-32}-4q^{-34}+460q^{-36}-713q^{-38}+662q^{-40}-333q^{-42}-116q^{-44}+492q^{-46}-624q^{-48}+450q^{-50}-37q^{-52}-434q^{-54}+775q^{-56}-811q^{-58}+524q^{-60}+q^{-62}-577q^{-64}+976q^{-66}-1060q^{-68}+779q^{-70}-227q^{-72}-388q^{-74}+840q^{-76}-967q^{-78}+728q^{-80}-250q^{-82}-270q^{-84}+594q^{-86}-624q^{-88}+356q^{-90}+62q^{-92}-428q^{-94}+588q^{-96}-465q^{-98}+124q^{-100}+274q^{-102}-578q^{-104}+662q^{-106}-511q^{-108}+211q^{-110}+135q^{-112}-399q^{-114}+517q^{-116}-473q^{-118}+312q^{-120}-101q^{-122}-88q^{-124}+207q^{-126}-253q^{-128}+225q^{-130}-153q^{-132}+72q^{-134}+6q^{-136}-54q^{-138}+72q^{-140}-73q^{-142}+54q^{-144}-31q^{-146}+12q^{-148}+4q^{-150}-10q^{-152}+11q^{-154}-10q^{-156}+6q^{-158}-3q^{-160}+q^{-162}$

Further Quantum Invariants

K11a127 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["K11a127"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-6t^{3}+17t^{2}-28t+33-28t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$

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]=

{ 137, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{10}-4q^{9}+9q^{8}-15q^{7}+19q^{6}-22q^{5}+22q^{4}-18q^{3}+14q^{2}-8q+4-q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-4}+2z^{10}a^{-6}+5z^{9}a^{-3}+13z^{9}a^{-5}+8z^{9}a^{-7}+4z^{8}a^{-2}+10z^{8}a^{-4}+20z^{8}a^{-6}+14z^{8}a^{-8}+z^{7}a^{-1}-12z^{7}a^{-3}-25z^{7}a^{-5}+2z^{7}a^{-7}+14z^{7}a^{-9}-14z^{6}a^{-2}-48z^{6}a^{-4}-64z^{6}a^{-6}-21z^{6}a^{-8}+9z^{6}a^{-10}-3z^{5}a^{-1}+z^{5}a^{-3}-8z^{5}a^{-5}-36z^{5}a^{-7}-20z^{5}a^{-9}+4z^{5}a^{-11}+16z^{4}a^{-2}+53z^{4}a^{-4}+54z^{4}a^{-6}+9z^{4}a^{-8}-7z^{4}a^{-10}+z^{4}a^{-12}+3z^{3}a^{-1}+11z^{3}a^{-3}+27z^{3}a^{-5}+32z^{3}a^{-7}+12z^{3}a^{-9}-z^{3}a^{-11}-6z^{2}a^{-2}-21z^{2}a^{-4}-20z^{2}a^{-6}-3z^{2}a^{-8}+2z^{2}a^{-10}-za^{-1}-4za^{-3}-10za^{-5}-10za^{-7}-3za^{-9}+4a^{-4}+4a^{-6}+a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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["K11a127"];

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]=

{ $t^{4}-6t^{3}+17t^{2}-28t+33-28t^{-1}+17t^{-2}-6t^{-3}+t^{-4}$, $q^{10}-4q^{9}+9q^{8}-15q^{7}+19q^{6}-22q^{5}+22q^{4}-18q^{3}+14q^{2}-8q+4-q^{-1}$ }

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]=

{}

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 $8$ $16$ $32$ ${\frac {124}{3}}$ ${\frac {20}{3}}$ $128$ ${\frac {448}{3}}$ ${\frac {160}{3}}$ $16$ ${\frac {256}{3}}$ $128$ ${\frac {992}{3}}$ ${\frac {160}{3}}$ ${\frac {7471}{15}}$ ${\frac {356}{15}}$ ${\frac {13324}{45}}$ $-{\frac {127}{9}}$ ${\frac {991}{15}}$

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=$4 is the signature of K11a127. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 21                       1 1 19                     3   -3 17                   6 1   5 15                 9 3     -6 13               10 6       4 11             12 9         -3 9           10 10           0 7         8 12             4 5       6 10               -4 3     3 9                 6 1   1 5                   -4 -1   3                     3 -3 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a126 K11a128