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

K11a179 K11a181

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 16 18 20 2 6 22 10 8

Alexander Polynomial:

t-4 - 5t-3 + 11t-2 - 17t-1 + 21 - 17t + 11t2 - 5t3 + t4

Conway Polynomial:

1 - 2z2 + z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{89, 0}

Jones Polynomial:

q-6 - 3q-5 + 6q-4 - 9q-3 + 12q-2 - 14q-1 + 14 - 12q + 9q2 - 5q3 + 3q4 - q5

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

{...}

A2 (sl(3)) Invariant:

q-18 + q-12 - 2q-10 + 2q-8 - q-6 - q-4 + q-2 - 3 + 3q2 - q4 + 2q6 + 2q8 - q10 + q12 - q14

HOMFLY-PT Polynomial:

- 4a-2z2 - 4a-2z4 - a-2z6 + 3 + 11z2 + 13z4 + 6z6 + z8 - 4a2 - 12a2z2 - 9a2z4 - 2a2z6 + 2a4 + 3a4z2 + a4z4

Kauffman Polynomial:

- 2a-5z3 + a-5z5 + 3a-4z2 - 7a-4z4 + 3a-4z6 - a-3z + 4a-3z3 - 8a-3z5 + 4a-3z7 + 4a-2z4 - 7a-2z6 + 4a-2z8 - 2a-1z + 5a-1z3 + a-1z5 - 4a-1z7 + 3a-1z9 + 3 - 16z2 + 31z4 - 20z6 + 5z8 + z10 - 6az3 + 18az5 - 16az7 + 6az9 + 4a2 - 20a2z2 + 31a2z4 - 22a2z6 + 5a2z8 + a2z10 + a3z3 - a3z5 - 5a3z7 + 3a3z9 + 2a4 - 5a4z2 + 8a4z4 - 11a4z6 + 4a4z8 - a5z + 6a5z3 - 9a5z5 + 3a5z7 + 2a6z2 - 3a6z4 + a6z6

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

{-2, 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=0 is the signature of 11180. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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 = 11                       1 j = 9                     2   j = 7                   3 1   j = 5                 6 2     j = 3               6 3       j = 1             8 6         j = -1           7 7           j = -3         5 7             j = -5       4 7               j = -7     2 5                 j = -9   1 4                   j = -11   2                     j = -13 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, 180]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 180]]
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, 9, 19, 10], X[20, 11, 21, 12], X[2, 13, 3, 14], X[6, 16, 7, 15], > X[22, 18, 1, 17], X[10, 19, 11, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 180]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -8, 4, -11, 5, -10, 6, -2, 7, -3, 8, -4, 9, -5, 10, > -6, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 180]]
Out[5]=	DTCode[4, 12, 14, 16, 18, 20, 2, 6, 22, 10, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 180]][t]
Out[6]=	-4 5 11 17 2 3 4 21 + t - -- + -- - -- - 17 t + 11 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 180]][z]
Out[7]=	2 4 6 8 1 - 2 z + z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 180]}
In[9]:=	{KnotDet[Knot[11, Alternating, 180]], KnotSignature[Knot[11, Alternating, 180]]}
Out[9]=	{89, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 180]][q]
Out[10]=	-6 3 6 9 12 14 2 3 4 5 14 + q - -- + -- - -- + -- - -- - 12 q + 9 q - 5 q + 3 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 180]}
In[12]:=	A2Invariant[Knot[11, Alternating, 180]][q]
Out[12]=	-18 -12 2 2 -6 -4 -2 2 4 6 8 10 -3 + q + q - --- + -- - q - q + q + 3 q - q + 2 q + 2 q - q + 10 8 q q 12 14 > q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 180]][a, z]
Out[13]=	2 4 2 4 2 4 z 2 2 4 2 4 4 z 2 4 3 - 4 a + 2 a + 11 z - ---- - 12 a z + 3 a z + 13 z - ---- - 9 a z + 2 2 a a 6 4 4 6 z 2 6 8 > a z + 6 z - -- - 2 a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 180]][a, z]
Out[14]=	2 2 4 z 2 z 5 2 3 z 2 2 4 2 3 + 4 a + 2 a - -- - --- - a z - 16 z + ---- - 20 a z - 5 a z + 3 a 4 a a 3 3 3 4 6 2 2 z 4 z 5 z 3 3 3 5 3 4 7 z > 2 a z - ---- + ---- + ---- - 6 a z + a z + 6 a z + 31 z - ---- + 5 3 a 4 a a a 4 5 5 5 4 z 2 4 4 4 6 4 z 8 z z 5 3 5 > ---- + 31 a z + 8 a z - 3 a z + -- - ---- + -- + 18 a z - a z - 2 5 3 a a a a 6 6 7 7 5 5 6 3 z 7 z 2 6 4 6 6 6 4 z 4 z > 9 a z - 20 z + ---- - ---- - 22 a z - 11 a z + a z + ---- - ---- - 4 2 3 a a a a 8 9 7 3 7 5 7 8 4 z 2 8 4 8 3 z > 16 a z - 5 a z + 3 a z + 5 z + ---- + 5 a z + 4 a z + ---- + 2 a a 9 3 9 10 2 10 > 6 a z + 3 a z + z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 180]], Vassiliev[3][Knot[11, Alternating, 180]]}
Out[15]=	{-2, 2}
In[16]:=	Kh[Knot[11, Alternating, 180]][q, t]
Out[16]=	7 1 2 1 4 2 5 4 7 - + 8 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 5 7 7 3 3 2 5 2 5 3 > ----- + ---- + --- + 6 q t + 6 q t + 3 q t + 6 q t + 2 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a180

K11a179 K11a181