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

L11a314 L11a316

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a315 Visit L11a315's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-29/2 - 3q-27/2 + 6q-25/2 - 11q-23/2 + 14q-21/2 - 15q-19/2 + 15q-17/2 - 13q-15/2 + 9q-13/2 - 6q-11/2 + 2q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- q-44 - q-42 + q-40 - q-38 + 3q-36 + 2q-30 - 3q-28 + 4q-26 - q-24 + 2q-22 + 2q-20 - q-18 + 3q-16 - q-14 + q-12

HOMFLY-PT Polynomial:

- 4a7z - 8a7z3 - 5a7z5 - a7z7 - 2a9z-1 - 10a9z - 16a9z3 - 10a9z5 - 2a9z7 + 3a11z-1 + 13a11z + 12a11z3 + 3a11z5 - a13z-1 - 3a13z - a13z3

Kauffman Polynomial:

4a7z - 8a7z3 + 5a7z5 - a7z7 - 6a8z4 + 7a8z6 - 2a8z8 + 2a9z-1 - 11a9z + 17a9z3 - 16a9z5 + 11a9z7 - 3a9z9 - 3a10 + 17a10z2 - 32a10z4 + 26a10z6 - 5a10z8 - a10z10 + 3a11z-1 - 16a11z + 24a11z3 - 20a11z5 + 19a11z7 - 7a11z9 - 3a12 + 22a12z2 - 39a12z4 + 35a12z6 - 10a12z8 - a12z10 + a13z-1 - a13z - 9a13z3 + 13a13z5 - 4a13z9 - a14 + 3a14z2 - 8a14z4 + 11a14z6 - 7a14z8 - a15z - 5a15z3 + 9a15z5 - 7a15z7 - a16z2 + 4a16z4 - 5a16z6 - a17z + 3a17z3 - 3a17z5 + a18z2 - a18z4

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -6                       1 j = -8                     2 1 j = -10                   4     j = -12                 5 2     j = -14               8 4       j = -16             7 5         j = -18           8 8           j = -20         6 7             j = -22       5 8               j = -24     2 7                 j = -26   1 4                   j = -28   2                     j = -30 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, 315]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 315]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[22, 17, 9, 18], > X[20, 9, 21, 10], X[6, 13, 7, 14], X[14, 7, 15, 8], X[8, 15, 1, 16], > X[4, 19, 5, 20], X[18, 5, 19, 6], X[16, 21, 17, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -9, 10, -6, 7, -8}, > {5, -1, 2, -3, 6, -7, 8, -11, 4, -10, 9, -5, 11, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(29/2) 3 6 11 14 15 15 13 9 q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 27/2 25/2 23/2 21/2 19/2 17/2 15/2 13/2 q q q q q q q q 6 2 -(7/2) > ----- + ---- - q 11/2 9/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-44 -42 -40 -38 3 2 3 4 -24 2 2 -18 -q - q + q - q + --- + --- - --- + --- - q + --- + --- - q + 36 30 28 26 22 20 q q q q q q 3 -14 -12 > --- - q + q 16 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 315]][a, z]
Out[8]=	9 11 13 -2 a 3 a a 7 9 11 13 7 3 ----- + ----- - --- - 4 a z - 10 a z + 13 a z - 3 a z - 8 a z - z z z 9 3 11 3 13 3 7 5 9 5 11 5 7 7 > 16 a z + 12 a z - a z - 5 a z - 10 a z + 3 a z - a z - 9 7 > 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 315]][a, z]
Out[9]=	9 11 13 10 12 14 2 a 3 a a 7 9 11 -3 a - 3 a - a + ---- + ----- + --- + 4 a z - 11 a z - 16 a z - z z z 13 15 17 10 2 12 2 14 2 16 2 > a z - a z - a z + 17 a z + 22 a z + 3 a z - a z + 18 2 7 3 9 3 11 3 13 3 15 3 17 3 > a z - 8 a z + 17 a z + 24 a z - 9 a z - 5 a z + 3 a z - 8 4 10 4 12 4 14 4 16 4 18 4 7 5 > 6 a z - 32 a z - 39 a z - 8 a z + 4 a z - a z + 5 a z - 9 5 11 5 13 5 15 5 17 5 8 6 > 16 a z - 20 a z + 13 a z + 9 a z - 3 a z + 7 a z + 10 6 12 6 14 6 16 6 7 7 9 7 > 26 a z + 35 a z + 11 a z - 5 a z - a z + 11 a z + 11 7 15 7 8 8 10 8 12 8 14 8 > 19 a z - 7 a z - 2 a z - 5 a z - 10 a z - 7 a z - 9 9 11 9 13 9 10 10 12 10 > 3 a z - 7 a z - 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 2 1 4 2 7 5 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 30 11 28 10 26 10 26 9 24 9 24 8 22 8 q t q t q t q t q t q t q t 8 6 7 8 8 7 5 8 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 22 7 20 7 20 6 18 6 18 5 16 5 16 4 14 4 q t q t q t q t q t q t q t q t 4 5 2 4 2 > ------ + ------ + ------ + ------ + ---- 14 3 12 3 12 2 10 2 8 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a315

L11a314 L11a316