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

K11n180 K11n182

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X3,13,4,12 X5,17,6,16 X14,8,15,7 X9,21,10,20 X11,19,12,18 X13,3,14,2 X22,16,1,15 X17,5,18,4 X19,11,20,10 X21,9,22,8

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

5t-2 - 11t-1 + 13 - 11t + 5t2

Conway Polynomial:

1 + 9z2 + 5z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{45, 4}

Jones Polynomial:

q2 - 2q3 + 4q4 - 6q5 + 8q6 - 7q7 + 7q8 - 5q9 + 3q10 - 2q11

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

{...}

A2 (sl(3)) Invariant:

q6 - q8 + q10 - q14 + 2q16 + 2q20 + 2q22 + q24 + 2q26 - 2q28 - q30 - q32 - 2q34

HOMFLY-PT Polynomial:

- 3a-10 - 2a-10z2 + 3a-8 + 5a-8z2 + 2a-8z4 + a-6 + 4a-6z2 + 2a-6z4 + 2a-4z2 + a-4z4

Kauffman Polynomial:

7a-13z - 10a-13z3 + 3a-13z5 - a-12z2 - 2a-12z6 + a-12z8 + 5a-11z - 12a-11z3 + 7a-11z5 - 3a-11z7 + a-11z9 + 3a-10 - 19a-10z2 + 25a-10z4 - 15a-10z6 + 4a-10z8 - 2a-9z + 7a-9z3 - 4a-9z5 + a-9z9 + 3a-8 - 11a-8z2 + 18a-8z4 - 10a-8z6 + 3a-8z8 + 6a-7z3 - 6a-7z5 + 3a-7z7 - a-6 + 5a-6z2 - 6a-6z4 + 3a-6z6 - 3a-5z3 + 2a-5z5 - 2a-4z2 + a-4z4

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

{9, 26}

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=4 is the signature of 11181. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                   2 j = 21                 1   j = 19               4 2   j = 17             3 1     j = 15           4 4       j = 13         4 3         j = 11       2 4           j = 9     2 4             j = 7     2               j = 5 1 2                 j = 3 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, 181]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 181]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 13, 4, 12], X[5, 17, 6, 16], X[14, 8, 15, 7], > X[9, 21, 10, 20], X[11, 19, 12, 18], X[13, 3, 14, 2], X[22, 16, 1, 15], > X[17, 5, 18, 4], X[19, 11, 20, 10], X[21, 9, 22, 8]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 181]]
Out[4]=	GaussCode[1, 7, -2, 9, -3, -1, 4, 11, -5, 10, -6, 2, -7, -4, 8, 3, -9, 6, -10, > 5, -11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 181]]
Out[5]=	DTCode[6, -12, -16, 14, -20, -18, -2, 22, -4, -10, -8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 181]][t]
Out[6]=	5 11 2 13 + -- - -- - 11 t + 5 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 181]][z]
Out[7]=	2 4 1 + 9 z + 5 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 181]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 181]], KnotSignature[Knot[11, NonAlternating, 181]]}
Out[9]=	{45, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 181]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 q - 2 q + 4 q - 6 q + 8 q - 7 q + 7 q - 5 q + 3 q - 2 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 181]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 181]][q]
Out[12]=	6 8 10 14 16 20 22 24 26 28 30 32 q - q + q - q + 2 q + 2 q + 2 q + q + 2 q - 2 q - q - q - 34 > 2 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 181]][a, z]
Out[13]=	2 2 2 2 4 4 4 -3 3 -6 2 z 5 z 4 z 2 z 2 z 2 z z --- + -- + a - ---- + ---- + ---- + ---- + ---- + ---- + -- 10 8 10 8 6 4 8 6 4 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 181]][a, z]
Out[14]=	2 2 2 2 2 3 3 3 -6 7 z 5 z 2 z z 19 z 11 z 5 z 2 z 10 z --- + -- - a + --- + --- - --- - --- - ----- - ----- + ---- - ---- - ----- - 10 8 13 11 9 12 10 8 6 4 13 a a a a a a a a a a a 3 3 3 3 4 4 4 4 5 5 12 z 7 z 6 z 3 z 25 z 18 z 6 z z 3 z 7 z > ----- + ---- + ---- - ---- + ----- + ----- - ---- + -- + ---- + ---- - 11 9 7 5 10 8 6 4 13 11 a a a a a a a a a a 5 5 5 6 6 6 6 7 7 8 4 z 6 z 2 z 2 z 15 z 10 z 3 z 3 z 3 z z > ---- - ---- + ---- - ---- - ----- - ----- + ---- - ---- + ---- + --- + 9 7 5 12 10 8 6 11 7 12 a a a a a a a a a a 8 8 9 9 4 z 3 z z z > ---- + ---- + --- + -- 10 8 11 9 a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 181]], Vassiliev[3][Knot[11, NonAlternating, 181]]}
Out[15]=	{9, 26}
In[16]:=	Kh[Knot[11, NonAlternating, 181]][q, t]
Out[16]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 2 q t + 2 q t + 2 q t + 4 q t + 2 q t + 4 q t + 13 4 13 5 15 5 15 6 17 6 17 7 19 7 > 4 q t + 3 q t + 4 q t + 4 q t + 3 q t + q t + 4 q t + 19 8 21 8 23 9 > 2 q t + q t + 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n181

K11n180 K11n182