© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a286 K11a288

Knotscape

This page is passe. Go here instead!    The Knot K11a287 Visit K11a287's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6271 X10,3,11,4 X16,6,17,5 X18,7,19,8 X20,10,21,9 X2,11,3,12 X8,13,9,14 X4,16,5,15 X22,17,1,18 X12,20,13,19 X14,21,15,22

Gauss Code:

{1, -6, 2, -8, 3, -1, 4, -7, 5, -2, 6, -10, 7, -11, 8, -3, 9, -4, 10, -5, 11, -9}

DT (Dowker-Thistlethwaite) Code:

6 10 16 18 20 2 8 4 22 12 14

Alexander Polynomial:

t-4 - 7t-3 + 21t-2 - 38t-1 + 47 - 38t + 21t2 - 7t3 + t4

Conway Polynomial:

1 - z2 - z4 + z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{181, 0}

Jones Polynomial:

q-6 - 4q-5 + 10q-4 - 18q-3 + 25q-2 - 29q-1 + 30 - 26q + 20q2 - 12q3 + 5q4 - q5

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q-18 - q-16 + 3q-12 - 5q-10 + 4q-8 - 2q-6 - 2q-4 + 5q-2 - 5 + 7q2 - 4q4 + q6 + 3q8 - 4q10 + 3q12 - q14

HOMFLY-PT Polynomial:

- a-2z2 - 2a-2z4 - a-2z6 + 2 + 4z2 + 6z4 + 4z6 + z8 - 2a2 - 6a2z2 - 6a2z4 - 2a2z6 + a4 + 2a4z2 + a4z4

Kauffman Polynomial:

a-5z5 - 3a-4z4 + 5a-4z6 + 4a-3z3 - 14a-3z5 + 12a-3z7 - 3a-2z2 + 13a-2z4 - 26a-2z6 + 17a-2z8 + 5a-1z3 - 9a-1z5 - 10a-1z7 + 13a-1z9 + 2 - 12z2 + 43z4 - 61z6 + 23z8 + 4z10 - az3 + 16az5 - 39az7 + 22az9 + 2a2 - 15a2z2 + 41a2z4 - 47a2z6 + 14a2z8 + 4a2z10 - a3z + 3a3z3 + 2a3z5 - 13a3z7 + 9a3z9 + a4 - 5a4z2 + 12a4z4 - 16a4z6 + 8a4z8 - a5z + 5a5z3 - 8a5z5 + 4a5z7 + a6z2 - 2a6z4 + a6z6

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-1, 1}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=0 is the signature of 11287. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     4   j = 7                   8 1   j = 5                 12 4     j = 3               14 8       j = 1             16 12         j = -1           14 15           j = -3         11 15             j = -5       7 14               j = -7     3 11                 j = -9   1 7                   j = -11   3                     j = -13 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 287]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 287]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[16, 6, 17, 5], X[18, 7, 19, 8], > X[20, 10, 21, 9], X[2, 11, 3, 12], X[8, 13, 9, 14], X[4, 16, 5, 15], > X[22, 17, 1, 18], X[12, 20, 13, 19], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 287]]
Out[4]=	GaussCode[1, -6, 2, -8, 3, -1, 4, -7, 5, -2, 6, -10, 7, -11, 8, -3, 9, -4, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 287]]
Out[5]=	DTCode[6, 10, 16, 18, 20, 2, 8, 4, 22, 12, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 287]][t]
Out[6]=	-4 7 21 38 2 3 4 47 + t - -- + -- - -- - 38 t + 21 t - 7 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 287]][z]
Out[7]=	2 4 6 8 1 - z - z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 287]}
In[9]:=	{KnotDet[Knot[11, Alternating, 287]], KnotSignature[Knot[11, Alternating, 287]]}
Out[9]=	{181, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 287]][q]
Out[10]=	-6 4 10 18 25 29 2 3 4 5 30 + q - -- + -- - -- + -- - -- - 26 q + 20 q - 12 q + 5 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 287]}
In[12]:=	A2Invariant[Knot[11, Alternating, 287]][q]
Out[12]=	-18 -16 3 5 4 2 2 5 2 4 6 8 -5 + q - q + --- - --- + -- - -- - -- + -- + 7 q - 4 q + q + 3 q - 12 10 8 6 4 2 q q q q q q 10 12 14 > 4 q + 3 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 287]][a, z]
Out[13]=	2 4 2 4 2 z 2 2 4 2 4 2 z 2 4 4 4 2 - 2 a + a + 4 z - -- - 6 a z + 2 a z + 6 z - ---- - 6 a z + a z + 2 2 a a 6 6 z 2 6 8 > 4 z - -- - 2 a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 287]][a, z]
Out[14]=	2 2 4 3 5 2 3 z 2 2 4 2 6 2 2 + 2 a + a - a z - a z - 12 z - ---- - 15 a z - 5 a z + a z + 2 a 3 3 4 4 4 z 5 z 3 3 3 5 3 4 3 z 13 z 2 4 > ---- + ---- - a z + 3 a z + 5 a z + 43 z - ---- + ----- + 41 a z + 3 a 4 2 a a a 5 5 5 4 4 6 4 z 14 z 9 z 5 3 5 5 5 > 12 a z - 2 a z + -- - ----- - ---- + 16 a z + 2 a z - 8 a z - 5 3 a a a 6 6 7 7 6 5 z 26 z 2 6 4 6 6 6 12 z 10 z > 61 z + ---- - ----- - 47 a z - 16 a z + a z + ----- - ----- - 4 2 3 a a a a 8 9 7 3 7 5 7 8 17 z 2 8 4 8 13 z > 39 a z - 13 a z + 4 a z + 23 z + ----- + 14 a z + 8 a z + ----- + 2 a a 9 3 9 10 2 10 > 22 a z + 9 a z + 4 z + 4 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 287]], Vassiliev[3][Knot[11, Alternating, 287]]}
Out[15]=	{-1, 1}
In[16]:=	Kh[Knot[11, Alternating, 287]][q, t]
Out[16]=	15 1 3 1 7 3 11 7 14 -- + 16 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 11 15 14 3 3 2 5 2 5 3 > ----- + ---- + --- + 12 q t + 14 q t + 8 q t + 12 q t + 4 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 8 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a287

K11a286 K11a288