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

K11a12 K11a14

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X16,7,17,8 X2,9,3,10 X22,12,1,11 X20,14,21,13 X18,16,19,15 X6,17,7,18 X14,20,15,19 X12,22,13,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 10 16 2 22 20 18 6 14 12

Alexander Polynomial:

3t-2 - 15t-1 + 25 - 15t + 3t2

Conway Polynomial:

1 - 3z2 + 3z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{61, 0}

Jones Polynomial:

q-4 - 2q-3 + 4q-2 - 6q-1 + 8 - 9q + 9q2 - 8q3 + 6q4 - 4q5 + 3q6 - q7

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

{...}

A2 (sl(3)) Invariant:

q-14 + q-12 - q-10 + q-8 - 2q-4 + 2q-2 - q6 + q8 - q10 + 2q14 - q16 + q18 + q20 - q22

HOMFLY-PT Polynomial:

- a-6z2 + a-4 + a-4z2 + a-4z4 - a-2 - a-2z2 + a-2z4 + 1 + z4 - a2 - 2a2z2 + a4

Kauffman Polynomial:

3a-7z3 - 4a-7z5 + a-7z7 - 5a-6z2 + 17a-6z4 - 14a-6z6 + 3a-6z8 + a-5z - 6a-5z3 + 15a-5z5 - 13a-5z7 + 3a-5z9 + a-4 - 12a-4z2 + 27a-4z4 - 18a-4z6 + a-4z8 + a-4z10 + 4a-3z - 18a-3z3 + 29a-3z5 - 21a-3z7 + 5a-3z9 + a-2 - 8a-2z2 + 13a-2z4 - 8a-2z6 + a-2z10 + 3a-1z - 8a-1z3 + 8a-1z5 - 5a-1z7 + 2a-1z9 + 1 - 2z2 + 2z4 - 2z6 + 2z8 + az - 2az3 + 2az7 + a2 - 3a2z2 + 2a2z6 + a3z - 3a3z3 + 2a3z5 + a4 - 2a4z2 + a4z4

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

{-3, 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 1113. 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                     2   j = 11                   2 1   j = 9                 4 2     j = 7               4 2       j = 5             5 4         j = 3           4 4           j = 1         4 5             j = -1       3 5               j = -3     1 3                 j = -5   1 3                   j = -7   1                     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, 13]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 13]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[16, 7, 17, 8], > X[2, 9, 3, 10], X[22, 12, 1, 11], X[20, 14, 21, 13], X[18, 16, 19, 15], > X[6, 17, 7, 18], X[14, 20, 15, 19], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 13]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -2, 5, -3, 6, -11, 7, -10, 8, -4, 9, -8, 10, > -7, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 13]]
Out[5]=	DTCode[4, 8, 10, 16, 2, 22, 20, 18, 6, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 13]][t]
Out[6]=	3 15 2 25 + -- - -- - 15 t + 3 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 13]][z]
Out[7]=	2 4 1 - 3 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 13]}
In[9]:=	{KnotDet[Knot[11, Alternating, 13]], KnotSignature[Knot[11, Alternating, 13]]}
Out[9]=	{61, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 13]][q]
Out[10]=	-4 2 4 6 2 3 4 5 6 7 8 + q - -- + -- - - - 9 q + 9 q - 8 q + 6 q - 4 q + 3 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, 13]}
In[12]:=	A2Invariant[Knot[11, Alternating, 13]][q]
Out[12]=	-14 -12 -10 -8 2 2 6 8 10 14 16 18 20 q + q - q + q - -- + -- - q + q - q + 2 q - q + q + q - 4 2 q q 22 > q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 13]][a, z]
Out[13]=	2 2 2 4 4 -4 -2 2 4 z z z 2 2 4 z z 1 + a - a - a + a - -- + -- - -- - 2 a z + z + -- + -- 6 4 2 4 2 a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 13]][a, z]
Out[14]=	2 2 -4 -2 2 4 z 4 z 3 z 3 2 5 z 12 z 1 + a + a + a + a + -- + --- + --- + a z + a z - 2 z - ---- - ----- - 5 3 a 6 4 a a a a 2 3 3 3 3 8 z 2 2 4 2 3 z 6 z 18 z 8 z 3 3 3 > ---- - 3 a z - 2 a z + ---- - ---- - ----- - ---- - 2 a z - 3 a z + 2 7 5 3 a a a a a 4 4 4 5 5 5 5 4 17 z 27 z 13 z 4 4 4 z 15 z 29 z 8 z > 2 z + ----- + ----- + ----- + a z - ---- + ----- + ----- + ---- + 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 7 3 5 6 14 z 18 z 8 z 2 6 z 13 z 21 z > 2 a z - 2 z - ----- - ----- - ---- + 2 a z + -- - ----- - ----- - 6 4 2 7 5 3 a a a a a a 7 8 8 9 9 9 10 10 5 z 7 8 3 z z 3 z 5 z 2 z z z > ---- + 2 a z + 2 z + ---- + -- + ---- + ---- + ---- + --- + --- a 6 4 5 3 a 4 2 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 13]], Vassiliev[3][Knot[11, Alternating, 13]]}
Out[15]=	{-3, 0}
In[16]:=	Kh[Knot[11, Alternating, 13]][q, t]
Out[16]=	5 1 1 1 3 1 3 3 3 - + 4 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 5 q t + 4 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 2 5 2 5 3 7 3 7 4 9 4 9 5 > 4 q t + 5 q t + 4 q t + 4 q t + 2 q t + 4 q t + 2 q t + 11 5 11 6 13 6 15 7 > 2 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a13

K11a12 K11a14