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

K11n87 K11n89

Knotscape

This page is passe. Go here instead!    The Knot K11n88 Visit K11n88's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X12,6,13,5 X16,8,17,7 X18,10,19,9 X2,11,3,12 X13,21,14,20 X6,16,7,15 X8,18,9,17 X19,1,20,22 X21,15,22,14

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 16 18 2 -20 6 8 -22 -14

Alexander Polynomial:

- t-4 + 3t-3 - 2t-2 + 1 - 2t2 + 3t3 - t4

Conway Polynomial:

1 + 3z2 - 4z4 - 5z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{11, 6}

Jones Polynomial:

q - q2 + 2q3 - 2q4 + 2q5 - q6 + q7 - q8

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

{72, ...}

A2 (sl(3)) Invariant:

q4 + q6 + q8 + q10 + q18 - q20 - q24 + q32 - q34

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + 3a-8 + 9a-8z2 + 6a-8z4 + a-8z6 - 5a-6 - 15a-6z2 - 16a-6z4 - 7a-6z6 - a-6z8 + 4a-4 + 10a-4z2 + 6a-4z4 + a-4z6

Kauffman Polynomial:

- a-11z + a-10 - 2a-10z2 + 4a-9z3 - 5a-9z5 + a-9z7 + 3a-8 - 12a-8z2 + 20a-8z4 - 12a-8z6 + 2a-8z8 + 5a-7z5 - 5a-7z7 + a-7z9 + 5a-6 - 24a-6z2 + 36a-6z4 - 19a-6z6 + 3a-6z8 - a-5z - 4a-5z3 + 10a-5z5 - 6a-5z7 + a-5z9 + 4a-4 - 14a-4z2 + 16a-4z4 - 7a-4z6 + a-4z8

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

{3, 6}

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=6 is the signature of 1188. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 17               1 j = 15             1 1 j = 13           1 1   j = 11         1 1 1   j = 9       1 1       j = 7     1 1         j = 5   1 2           j = 3                 j = 1 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, NonAlternating, 88]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 88]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[16, 8, 17, 7], > X[18, 10, 19, 9], X[2, 11, 3, 12], X[13, 21, 14, 20], X[6, 16, 7, 15], > X[8, 18, 9, 17], X[19, 1, 20, 22], X[21, 15, 22, 14]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 88]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -8, 4, -9, 5, -2, 6, -3, -7, 11, 8, -4, 9, -5, -10, > 7, -11, 10]
In[5]:=	DTCode[Knot[11, NonAlternating, 88]]
Out[5]=	DTCode[4, 10, 12, 16, 18, 2, -20, 6, 8, -22, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 88]][t]
Out[6]=	-4 3 2 2 3 4 1 - t + -- - -- - 2 t + 3 t - t 3 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 88]][z]
Out[7]=	2 4 6 8 1 + 3 z - 4 z - 5 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 88]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 88]], KnotSignature[Knot[11, NonAlternating, 88]]}
Out[9]=	{11, 6}
In[10]:=	J=Jones[Knot[11, NonAlternating, 88]][q]
Out[10]=	2 3 4 5 6 7 8 q - q + 2 q - 2 q + 2 q - q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[7, 2], Knot[11, NonAlternating, 88]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 88]][q]
Out[12]=	4 6 8 10 18 20 24 32 34 q + q + q + q + q - q - q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 88]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -10 3 5 4 z 9 z 15 z 10 z 6 z 16 z 6 z z -a + -- - -- + -- - --- + ---- - ----- + ----- + ---- - ----- + ---- + -- - 8 6 4 10 8 6 4 8 6 4 8 a a a a a a a a a a a 6 6 8 7 z z z > ---- + -- - -- 6 4 6 a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 88]][a, z]
Out[14]=	2 2 2 2 3 3 -10 3 5 4 z z 2 z 12 z 24 z 14 z 4 z 4 z a + -- + -- + -- - --- - -- - ---- - ----- - ----- - ----- + ---- - ---- + 8 6 4 11 5 10 8 6 4 9 5 a a a a a a a a a a a 4 4 4 5 5 5 6 6 6 7 20 z 36 z 16 z 5 z 5 z 10 z 12 z 19 z 7 z z > ----- + ----- + ----- - ---- + ---- + ----- - ----- - ----- - ---- + -- - 8 6 4 9 7 5 8 6 4 9 a a a a a a a a a a 7 7 8 8 8 9 9 5 z 6 z 2 z 3 z z z z > ---- - ---- + ---- + ---- + -- + -- + -- 7 5 8 6 4 7 5 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 88]], Vassiliev[3][Knot[11, NonAlternating, 88]]}
Out[15]=	{3, 6}
In[16]:=	Kh[Knot[11, NonAlternating, 88]][q, t]
Out[16]=	5 5 7 q q 7 9 9 2 11 2 11 3 13 3 11 4 2 q + q + -- + -- + q t + q t + q t + q t + q t + q t + q t + 2 t t 13 4 15 4 15 5 17 5 > q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n88

K11n87 K11n89