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

K11a302 K11a304

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 18 22 16 4 8 20 12 2 14

Alexander Polynomial:

- 3t-3 + 15t-2 - 34t-1 + 45 - 34t + 15t2 - 3t3

Conway Polynomial:

1 - z2 - 3z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{149, 0}

Jones Polynomial:

q-4 - 4q-3 + 10q-2 - 16q-1 + 21 - 24q + 24q2 - 20q3 + 15q4 - 9q5 + 4q6 - q7

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

{...}

A2 (sl(3)) Invariant:

q-12 - 2q-10 + 2q-8 + 3q-6 - 4q-4 + 4q-2 - 3 - q2 + 2q4 - 3q6 + 5q8 - 3q10 + q12 + 3q14 - 3q16 + 2q18 - q22

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-7z + 3a-7z3 - 3a-7z5 + a-7z7 + a-6 - 6a-6z2 + 14a-6z4 - 13a-6z6 + 4a-6z8 - 2a-5z + 5a-5z3 + 8a-5z5 - 16a-5z7 + 6a-5z9 + 3a-4 - 20a-4z2 + 52a-4z4 - 45a-4z6 + 7a-4z8 + 3a-4z10 - 2a-3z + 5a-3z3 + 18a-3z5 - 41a-3z7 + 17a-3z9 + 2a-2 - 19a-2z2 + 58a-2z4 - 69a-2z6 + 20a-2z8 + 3a-2z10 - 3a-1z + 14a-1z3 - 19a-1z5 - 8a-1z7 + 11a-1z9 - z2 + 11z4 - 27z6 + 17z8 - 2az + 11az3 - 22az5 + 16az7 - a2 + 4a2z2 - 8a2z4 + 10a2z6 + 4a3z5 + a4z4

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

{-1, 1}

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 11303. 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 = 15                       1 j = 13                     3   j = 11                   6 1   j = 9                 9 3     j = 7               11 6       j = 5             13 9         j = 3           11 11           j = 1         10 13             j = -1       7 12               j = -3     3 9                 j = -5   1 7                   j = -7   3                     j = -9 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, 303]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 303]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[18, 5, 19, 6], X[22, 8, 1, 7], > X[16, 10, 17, 9], X[4, 11, 5, 12], X[8, 14, 9, 13], X[20, 16, 21, 15], > X[12, 18, 13, 17], X[2, 19, 3, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 303]]
Out[4]=	GaussCode[1, -10, 2, -6, 3, -1, 4, -7, 5, -2, 6, -9, 7, -11, 8, -5, 9, -3, 10, > -8, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 303]]
Out[5]=	DTCode[6, 10, 18, 22, 16, 4, 8, 20, 12, 2, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 303]][t]
Out[6]=	3 15 34 2 3 45 - -- + -- - -- - 34 t + 15 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 303]][z]
Out[7]=	2 4 6 1 - z - 3 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 303]}
In[9]:=	{KnotDet[Knot[11, Alternating, 303]], KnotSignature[Knot[11, Alternating, 303]]}
Out[9]=	{149, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 303]][q]
Out[10]=	-4 4 10 16 2 3 4 5 6 7 21 + q - -- + -- - -- - 24 q + 24 q - 20 q + 15 q - 9 q + 4 q - q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 303]}
In[12]:=	A2Invariant[Knot[11, Alternating, 303]][q]
Out[12]=	-12 2 2 3 4 4 2 4 6 8 10 12 -3 + q - --- + -- + -- - -- + -- - q + 2 q - 3 q + 5 q - 3 q + q + 10 8 6 4 2 q q q q q 14 16 18 22 > 3 q - 3 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 303]][a, z]
Out[13]=	2 2 2 4 4 -6 3 2 2 z 6 z 7 z 2 2 4 3 z 6 z 2 4 -a + -- - -- + a - -- + ---- - ---- + a z - z + ---- - ---- + a z - 4 2 6 4 2 4 2 a a a a a a a 6 6 2 z > z - ---- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 303]][a, z]
Out[14]=	2 2 2 -6 3 2 2 z 2 z 2 z 3 z 2 6 z 20 z 19 z a + -- + -- - a - -- - --- - --- - --- - 2 a z - z - ---- - ----- - ----- + 4 2 7 5 3 a 6 4 2 a a a a a a a a 3 3 3 3 4 4 2 2 3 z 5 z 5 z 14 z 3 4 14 z 52 z > 4 a z + ---- + ---- + ---- + ----- + 11 a z + 11 z + ----- + ----- + 7 5 3 a 6 4 a a a a a 4 5 5 5 5 58 z 2 4 4 4 3 z 8 z 18 z 19 z 5 3 5 > ----- - 8 a z + a z - ---- + ---- + ----- - ----- - 22 a z + 4 a z - 2 7 5 3 a a a a a 6 6 6 7 7 7 7 6 13 z 45 z 69 z 2 6 z 16 z 41 z 8 z > 27 z - ----- - ----- - ----- + 10 a z + -- - ----- - ----- - ---- + 6 4 2 7 5 3 a a a a a a a 8 8 8 9 9 9 10 10 7 8 4 z 7 z 20 z 6 z 17 z 11 z 3 z 3 z > 16 a z + 17 z + ---- + ---- + ----- + ---- + ----- + ----- + ----- + ----- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 303]], Vassiliev[3][Knot[11, Alternating, 303]]}
Out[15]=	{-1, 1}
In[16]:=	Kh[Knot[11, Alternating, 303]][q, t]
Out[16]=	12 1 3 1 7 3 9 7 -- + 10 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 13 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 > 11 q t + 11 q t + 13 q t + 9 q t + 11 q t + 6 q t + 9 q t + 9 5 11 5 11 6 13 6 15 7 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a303

K11a302 K11a304