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

K11a287 K11a289

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,3,11,4 X16,6,17,5 X18,7,19,8 X20,10,21,9 X4,11,5,12 X8,14,9,13 X2,16,3,15 X22,17,1,18 X14,19,15,20 X12,22,13,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 16 18 20 4 8 2 22 14 12

Alexander Polynomial:

t-4 - 7t-3 + 23t-2 - 44t-1 + 55 - 44t + 23t2 - 7t3 + t4

Conway Polynomial:

1 + z2 + z4 + z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{205, 0}

Jones Polynomial:

- q-5 + 5q-4 - 13q-3 + 22q-2 - 29q-1 + 34 - 33q + 29q2 - 21q3 + 12q4 - 5q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- q-14 + 3q-12 - 5q-10 + 3q-8 + q-6 - 5q-4 + 8q-2 - 5 + 6q2 - q4 - 2q6 + 5q8 - 6q10 + 3q12 - 2q16 + q18

HOMFLY-PT Polynomial:

a-4z2 + a-4z4 - a-2 - 4a-2z2 - 5a-2z4 - 2a-2z6 + 3 + 6z2 + 7z4 + 4z6 + z8 - a2 - 2a2z2 - 2a2z4 - a2z6

Kauffman Polynomial:

- a-6z4 + a-6z6 + 4a-5z3 - 8a-5z5 + 5a-5z7 - 3a-4z2 + 14a-4z4 - 21a-4z6 + 11a-4z8 - a-3z + 8a-3z3 - 4a-3z5 - 15a-3z7 + 12a-3z9 + a-2 - 12a-2z2 + 47a-2z4 - 65a-2z6 + 21a-2z8 + 5a-2z10 - 2a-1z + 8a-1z3 + 6a-1z5 - 45a-1z7 + 28a-1z9 + 3 - 15z2 + 53z4 - 79z6 + 30z8 + 5z10 - 2az + 10az3 - 13az5 - 12az7 + 16az9 + a2 - 6a2z2 + 19a2z4 - 31a2z6 + 20a2z8 - a3z + 6a3z3 - 14a3z5 + 13a3z7 - 2a4z4 + 5a4z6 + a5z5

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

{1, 0}

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 11288. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     4   j = 9                   8 1   j = 7                 13 4     j = 5               16 8       j = 3             17 13         j = 1           17 16           j = -1         13 18             j = -3       9 16               j = -5     4 13                 j = -7   1 9                   j = -9   4                     j = -11 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, 288]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 288]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[16, 6, 17, 5], X[18, 7, 19, 8], > X[20, 10, 21, 9], X[4, 11, 5, 12], X[8, 14, 9, 13], X[2, 16, 3, 15], > X[22, 17, 1, 18], X[14, 19, 15, 20], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 288]]
Out[4]=	GaussCode[1, -8, 2, -6, 3, -1, 4, -7, 5, -2, 6, -11, 7, -10, 8, -3, 9, -4, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 288]]
Out[5]=	DTCode[6, 10, 16, 18, 20, 4, 8, 2, 22, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 288]][t]
Out[6]=	-4 7 23 44 2 3 4 55 + t - -- + -- - -- - 44 t + 23 t - 7 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 288]][z]
Out[7]=	2 4 6 8 1 + z + z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 288]}
In[9]:=	{KnotDet[Knot[11, Alternating, 288]], KnotSignature[Knot[11, Alternating, 288]]}
Out[9]=	{205, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 288]][q]
Out[10]=	-5 5 13 22 29 2 3 4 5 6 34 - q + -- - -- + -- - -- - 33 q + 29 q - 21 q + 12 q - 5 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 288]}
In[12]:=	A2Invariant[Knot[11, Alternating, 288]][q]
Out[12]=	-14 3 5 3 -6 5 8 2 4 6 8 10 -5 - q + --- - --- + -- + q - -- + -- + 6 q - q - 2 q + 5 q - 6 q + 12 10 8 4 2 q q q q q 12 16 18 > 3 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 288]][a, z]
Out[13]=	2 2 4 4 -2 2 2 z 4 z 2 2 4 z 5 z 2 4 6 3 - a - a + 6 z + -- - ---- - 2 a z + 7 z + -- - ---- - 2 a z + 4 z - 4 2 4 2 a a a a 6 2 z 2 6 8 > ---- - a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 288]][a, z]
Out[14]=	2 2 -2 2 z 2 z 3 2 3 z 12 z 2 2 3 + a + a - -- - --- - 2 a z - a z - 15 z - ---- - ----- - 6 a z + 3 a 4 2 a a a 3 3 3 4 4 4 4 z 8 z 8 z 3 3 3 4 z 14 z 47 z > ---- + ---- + ---- + 10 a z + 6 a z + 53 z - -- + ----- + ----- + 5 3 a 6 4 2 a a a a a 5 5 5 2 4 4 4 8 z 4 z 6 z 5 3 5 5 5 > 19 a z - 2 a z - ---- - ---- + ---- - 13 a z - 14 a z + a z - 5 3 a a a 6 6 6 7 7 7 6 z 21 z 65 z 2 6 4 6 5 z 15 z 45 z > 79 z + -- - ----- - ----- - 31 a z + 5 a z + ---- - ----- - ----- - 6 4 2 5 3 a a a a a a 8 8 9 9 7 3 7 8 11 z 21 z 2 8 12 z 28 z > 12 a z + 13 a z + 30 z + ----- + ----- + 20 a z + ----- + ----- + 4 2 3 a a a a 10 9 10 5 z > 16 a z + 5 z + ----- 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 288]], Vassiliev[3][Knot[11, Alternating, 288]]}
Out[15]=	{1, 0}
In[16]:=	Kh[Knot[11, Alternating, 288]][q, t]
Out[16]=	18 1 4 1 9 4 13 9 16 -- + 17 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 13 3 3 2 5 2 5 3 7 3 > --- + 16 q t + 17 q t + 13 q t + 16 q t + 8 q t + 13 q t + q t 7 4 9 4 9 5 11 5 13 6 > 4 q t + 8 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a288

K11a287 K11a289