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

K11a180 K11a182

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X12,3,13,4 X14,6,15,5 X16,8,17,7 X18,10,19,9 X20,11,21,12 X2,13,3,14 X8,16,9,15 X6,18,7,17 X22,20,1,19 X10,21,11,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 16 18 20 2 8 6 22 10

Alexander Polynomial:

2t-3 - 11t-2 + 23t-1 - 27 + 23t - 11t2 + 2t3

Conway Polynomial:

1 - 3z2 + z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a102, K11a199, ...}

Determinant and Signature:

{99, 2}

Jones Polynomial:

q-3 - 3q-2 + 7q-1 - 10 + 14q - 16q2 + 15q3 - 14q4 + 10q5 - 5q6 + 3q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-10 - q-6 + 3q-4 + 1 + 2q2 - 4q4 + q6 - 3q8 + q12 - 2q14 + 4q16 + q22 - q24

HOMFLY-PT Polynomial:

- 2a-6z2 - a-6z4 + 2a-4 + 4a-4z2 + 3a-4z4 + a-4z6 - 3a-2 - 3a-2z2 + a-2z4 + a-2z6 + 1 - 3z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

- 2a-9z3 + a-9z5 + 4a-8z2 - 7a-8z4 + 3a-8z6 - 2a-7z + 4a-7z3 - 7a-7z5 + 4a-7z7 + 7a-6z2 - 3a-6z4 - 4a-6z6 + 4a-6z8 - 7a-5z + 12a-5z3 - 5a-5z5 - a-5z7 + 3a-5z9 + 2a-4 - 5a-4z2 + 15a-4z4 - 15a-4z6 + 6a-4z8 + a-4z10 - 5a-3z + 3a-3z3 + 5a-3z5 - 10a-3z7 + 6a-3z9 + 3a-2 - 10a-2z2 + 15a-2z4 - 17a-2z6 + 6a-2z8 + a-2z10 + 2a-1z3 - 6a-1z5 - 2a-1z7 + 3a-1z9 + 1 + z2 + z4 - 8z6 + 4z8 + 5az3 - 8az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

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

{-3, -2}

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=2 is the signature of 11181. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     2   j = 13                   3 1   j = 11                 7 2     j = 9               7 3       j = 7             8 7         j = 5           8 7           j = 3         6 8             j = 1       5 9               j = -1     2 5                 j = -3   1 5                   j = -5   2                     j = -7 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, Alternating, 181]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 181]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[14, 6, 15, 5], X[16, 8, 17, 7], > X[18, 10, 19, 9], X[20, 11, 21, 12], X[2, 13, 3, 14], X[8, 16, 9, 15], > X[6, 18, 7, 17], X[22, 20, 1, 19], X[10, 21, 11, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 181]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -9, 4, -8, 5, -11, 6, -2, 7, -3, 8, -4, 9, -5, 10, > -6, 11, -10]
In[5]:=	DTCode[Knot[11, Alternating, 181]]
Out[5]=	DTCode[4, 12, 14, 16, 18, 20, 2, 8, 6, 22, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 181]][t]
Out[6]=	2 11 23 2 3 -27 + -- - -- + -- + 23 t - 11 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 181]][z]
Out[7]=	2 4 6 1 - 3 z + z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 102], Knot[11, Alternating, 181], > Knot[11, Alternating, 199]}
In[9]:=	{KnotDet[Knot[11, Alternating, 181]], KnotSignature[Knot[11, Alternating, 181]]}
Out[9]=	{99, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 181]][q]
Out[10]=	-3 3 7 2 3 4 5 6 7 8 -10 + q - -- + - + 14 q - 16 q + 15 q - 14 q + 10 q - 5 q + 3 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 181]}
In[12]:=	A2Invariant[Knot[11, Alternating, 181]][q]
Out[12]=	-10 -6 3 2 4 6 8 12 14 16 22 24 1 + q - q + -- + 2 q - 4 q + q - 3 q + q - 2 q + 4 q + q - q 4 q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 181]][a, z]
Out[13]=	2 2 2 4 4 4 2 3 2 2 2 z 4 z 3 z 2 2 4 z 3 z z 1 + -- - -- + a - 3 z - ---- + ---- - ---- + a z - 2 z - -- + ---- + -- + 4 2 6 4 2 6 4 2 a a a a a a a a 6 6 z z > -- + -- 4 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 181]][a, z]
Out[14]=	2 2 2 2 2 3 2 2 z 7 z 5 z 2 4 z 7 z 5 z 10 z 1 + -- + -- - a - --- - --- - --- + z + ---- + ---- - ---- - ----- + 4 2 7 5 3 8 6 4 2 a a a a a a a a a 3 3 3 3 3 4 4 2 2 2 z 4 z 12 z 3 z 2 z 3 4 7 z 3 z > 3 a z - ---- + ---- + ----- + ---- + ---- + 5 a z + z - ---- - ---- + 9 7 5 3 a 8 6 a a a a a a 4 4 5 5 5 5 5 15 z 15 z 2 4 z 7 z 5 z 5 z 6 z 5 6 > ----- + ----- - 3 a z + -- - ---- - ---- + ---- - ---- - 8 a z - 8 z + 4 2 9 7 5 3 a a a a a a a 6 6 6 6 7 7 7 7 3 z 4 z 15 z 17 z 2 6 4 z z 10 z 2 z 7 > ---- - ---- - ----- - ----- + a z + ---- - -- - ----- - ---- + 3 a z + 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 9 9 9 10 10 8 4 z 6 z 6 z 3 z 6 z 3 z z z > 4 z + ---- + ---- + ---- + ---- + ---- + ---- + --- + --- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 181]], Vassiliev[3][Knot[11, Alternating, 181]]}
Out[15]=	{-3, -2}
In[16]:=	Kh[Knot[11, Alternating, 181]][q, t]
Out[16]=	3 1 2 1 5 2 5 5 q 3 9 q + 6 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 8 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 8 q t + 7 q t + 8 q t + 7 q t + 7 q t + 3 q t + 7 q t + 11 5 13 5 13 6 15 6 17 7 > 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a181

K11a180 K11a182