© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:

L11a37 L11a39

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a38 Visit L11a38's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,10,13,9 X18,13,19,14 X16,7,17,8 X8,17,9,18 X22,19,5,20 X20,16,21,15 X14,22,15,21 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 7q-9/2 + 13q-7/2 - 18q-5/2 + 20q-3/2 - 21q-1/2 + 18q1/2 - 14q3/2 + 8q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-20 + 3q-14 - 4q-12 + q-10 + q-8 - 3q-6 + 4q-4 - 3q-2 + 4 + q2 + 5q6 - 2q8 + q10 + q12 - q14

HOMFLY-PT Polynomial:

a-3z + a-3z3 - 2a-1z-1 - 5a-1z - 5a-1z3 - 2a-1z5 + 4az-1 + 10az + 9az3 + 4az5 + az7 - 3a3z-1 - 8a3z - 6a3z3 - 2a3z5 + a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

- a-4z2 + 2a-4z4 - a-4z6 + 2a-3z - 8a-3z3 + 10a-3z5 - 4a-3z7 - 5a-2z4 + 12a-2z6 - 6a-2z8 - 2a-1z-1 + 11a-1z - 28a-1z3 + 29a-1z5 - 4a-1z7 - 4a-1z9 + 2 + 3z2 - 23z4 + 37z6 - 15z8 - z10 - 4az-1 + 21az - 43az3 + 36az5 - az7 - 8az9 + 3a2 - 2a2z2 - 19a2z4 + 32a2z6 - 15a2z8 - a2z10 - 3a3z-1 + 15a3z - 28a3z3 + 24a3z5 - 6a3z7 - 4a3z9 + 3a4 - 7a4z2 + 2a4z4 + 5a4z6 - 6a4z8 - a5z-1 + 2a5z - 3a5z3 + 6a5z5 - 5a5z7 + a6 - 3a6z2 + 5a6z4 - 3a6z6 - a7z + 2a7z3 - a7z5

Khovanov Homology:

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 = 10                       1 j = 8                     3   j = 6                   5 1   j = 4                 9 3     j = 2               9 5       j = 0             12 9         j = -2           10 11           j = -4         8 10             j = -6       5 10               j = -8     2 8                 j = -10   1 5                   j = -12   2                     j = -14 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]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 38]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 38]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 10, 13, 9], X[18, 13, 19, 14], > X[16, 7, 17, 8], X[8, 17, 9, 18], X[22, 19, 5, 20], X[20, 16, 21, 15], > X[14, 22, 15, 21], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 5, -6, 3, -2, 11, -3, 4, -9, 8, -5, 6, -4, > 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 7 13 18 20 21 -q + ----- - ---- + ---- - ---- + ---- - ------- + 18 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 14 q + 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 3 4 -10 -8 3 4 3 2 6 8 10 4 + q + --- - --- + q + q - -- + -- - -- + q + 5 q - 2 q + q + 14 12 6 4 2 q q q q q 12 14 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 38]][a, z]
Out[8]=	3 5 3 3 -2 4 a 3 a a z 5 z 3 5 z 5 z --- + --- - ---- + -- + -- - --- + 10 a z - 8 a z + 2 a z + -- - ---- + a z z z z 3 a 3 a a a 5 3 3 3 5 3 2 z 5 3 5 7 > 9 a z - 6 a z + a z - ---- + 4 a z - 2 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 38]][a, z]
Out[9]=	3 5 2 4 6 2 4 a 3 a a 2 z 11 z 3 2 + 3 a + 3 a + a - --- - --- - ---- - -- + --- + ---- + 21 a z + 15 a z + a z z z z 3 a a 2 3 3 5 7 2 z 2 2 4 2 6 2 8 z 28 z > 2 a z - a z + 3 z - -- - 2 a z - 7 a z - 3 a z - ---- - ----- - 4 3 a a a 4 4 3 3 3 5 3 7 3 4 2 z 5 z 2 4 > 43 a z - 28 a z - 3 a z + 2 a z - 23 z + ---- - ---- - 19 a z + 4 2 a a 5 5 4 4 6 4 10 z 29 z 5 3 5 5 5 7 5 > 2 a z + 5 a z + ----- + ----- + 36 a z + 24 a z + 6 a z - a z + 3 a a 6 6 7 7 6 z 12 z 2 6 4 6 6 6 4 z 4 z 7 > 37 z - -- + ----- + 32 a z + 5 a z - 3 a z - ---- - ---- - a z - 4 2 3 a a a a 8 9 3 7 5 7 8 6 z 2 8 4 8 4 z 9 > 6 a z - 5 a z - 15 z - ---- - 15 a z - 6 a z - ---- - 8 a z - 2 a a 3 9 10 2 10 > 4 a z - z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 2 1 5 2 8 5 10 12 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 8 10 10 2 2 2 4 2 4 3 > ----- + ---- + ---- + 9 t + 9 q t + 5 q t + 9 q t + 3 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a38

L11a37 L11a39