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

L11a374 L11a376

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a375 Visit L11a375's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X12,1,13,2 X2,13,3,14 X14,3,15,4 X22,7,11,8 X18,10,19,9 X20,16,21,15 X16,6,17,5 X6,18,7,17 X10,20,1,19 X8,11,9,12 X4,21,5,22

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 7q-9/2 + 12q-7/2 - 16q-5/2 + 18q-3/2 - 19q-1/2 + 15q1/2 - 12q3/2 + 7q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 - q-16 + 3q-14 - 2q-12 + q-10 + q-8 - 3q-6 + 5q-4 - 2q-2 + 6 + q2 + 2q6 - 3q8 + q10 - q12

HOMFLY-PT Polynomial:

6a-1z + 9a-1z3 + 5a-1z5 + a-1z7 - az-1 - 13az - 24az3 - 19az5 - 7az7 - az9 + a3z-1 + 6a3z + 9a3z3 + 5a3z5 + a3z7

Kauffman Polynomial:

- 2a-4z2 + 3a-4z4 - a-4z6 + a-3z - 5a-3z3 + 8a-3z5 - 3a-3z7 + 4a-2z2 - 10a-2z4 + 13a-2z6 - 5a-2z8 - 6a-1z + 13a-1z3 - 13a-1z5 + 12a-1z7 - 5a-1z9 + 10z2 - 22z4 + 19z6 - 4z8 - 2z10 + az-1 - 14az + 32az3 - 38az5 + 27az7 - 10az9 - a2 + 10a2z2 - 21a2z4 + 17a2z6 - 5a2z8 - 2a2z10 + a3z-1 - 6a3z + 10a3z3 - 9a3z5 + 7a3z7 - 5a3z9 + 4a4z2 - 7a4z4 + 9a4z6 - 6a4z8 - 2a5z3 + 7a5z5 - 5a5z7 - 2a6z2 + 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                     2   j = 6                   5 1   j = 4                 7 2     j = 2               8 5       j = 0             11 7         j = -2           8 9           j = -4         8 10             j = -6       5 9               j = -8     2 7                 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, 375]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 375]]
Out[4]=	PD[X[12, 1, 13, 2], X[2, 13, 3, 14], X[14, 3, 15, 4], X[22, 7, 11, 8], > X[18, 10, 19, 9], X[20, 16, 21, 15], X[16, 6, 17, 5], X[6, 18, 7, 17], > X[10, 20, 1, 19], X[8, 11, 9, 12], X[4, 21, 5, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 7, -8, 4, -10, 5, -9}, > {10, -1, 2, -3, 6, -7, 8, -5, 9, -6, 11, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 7 12 16 18 19 -q + ----- - ---- + ---- - ---- + ---- - ------- + 15 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 > 12 q + 7 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 3 2 -10 -8 3 5 2 2 6 8 6 + q - q + --- - --- + q + q - -- + -- - -- + q + 2 q - 3 q + 14 12 6 4 2 q q q q q 10 12 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 375]][a, z]
Out[8]=	3 3 5 a a 6 z 3 9 z 3 3 3 5 z 5 -(-) + -- + --- - 13 a z + 6 a z + ---- - 24 a z + 9 a z + ---- - 19 a z + z z a a a 7 3 5 z 7 3 7 9 > 5 a z + -- - 7 a z + a z - a z a
In[9]:=	Kauffman[Link[11, Alternating, 375]][a, z]
Out[9]=	3 2 2 2 a a z 6 z 3 7 2 2 z 4 z -a + - + -- + -- - --- - 14 a z - 6 a z - a z + 10 z - ---- + ---- + z z 3 a 4 2 a a a 3 3 2 2 4 2 6 2 5 z 13 z 3 3 3 > 10 a z + 4 a z - 2 a z - ---- + ----- + 32 a z + 10 a z - 3 a a 4 4 5 3 7 3 4 3 z 10 z 2 4 4 4 6 4 > 2 a z + 2 a z - 22 z + ---- - ----- - 21 a z - 7 a z + 5 a z + 4 2 a a 5 5 6 6 8 z 13 z 5 3 5 5 5 7 5 6 z 13 z > ---- - ----- - 38 a z - 9 a z + 7 a z - a z + 19 z - -- + ----- + 3 a 4 2 a a a 7 7 2 6 4 6 6 6 3 z 12 z 7 3 7 5 7 > 17 a z + 9 a z - 3 a z - ---- + ----- + 27 a z + 7 a z - 5 a z - 3 a a 8 9 8 5 z 2 8 4 8 5 z 9 3 9 10 > 4 z - ---- - 5 a z - 6 a z - ---- - 10 a z - 5 a z - 2 z - 2 a a 2 10 > 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 2 1 5 2 7 5 9 11 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 8 2 2 2 4 2 4 3 > ----- + ---- + ---- + 7 t + 8 q t + 5 q t + 7 q t + 2 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 + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a375

L11a374 L11a376