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

K11a347 K11a349

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 18 16 12 4 2 20 22 10 8 14

Alexander Polynomial:

t-4 - 7t-3 + 19t-2 - 29t-1 + 33 - 29t + 19t2 - 7t3 + t4

Conway Polynomial:

1 - 3z4 + z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{145, 4}

Jones Polynomial:

- q-1 + 5 - 9q + 15q2 - 20q3 + 23q4 - 23q5 + 20q6 - 15q7 + 9q8 - 4q9 + q10

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

{...}

A2 (sl(3)) Invariant:

- q-2 + 3 - q2 + 3q4 + 2q6 - 3q8 + 4q10 - 6q12 + 2q14 - q16 - q18 + 4q20 - 3q22 + 2q24 - q26 - q28 + q30

HOMFLY-PT Polynomial:

2a-8z2 + a-8z4 + a-6 - 3a-6z2 - 6a-6z4 - 2a-6z6 - 3a-4 - a-4z2 + 4a-4z4 + 4a-4z6 + a-4z8 + 3a-2 + 2a-2z2 - 2a-2z4 - a-2z6

Kauffman Polynomial:

a-12z4 - a-11z3 + 4a-11z5 + 2a-10z2 - 7a-10z4 + 9a-10z6 - a-9z + 9a-9z3 - 19a-9z5 + 14a-9z7 - 3a-8z2 + 11a-8z4 - 24a-8z6 + 15a-8z8 - a-7z + 5a-7z3 - 5a-7z5 - 14a-7z7 + 11a-7z9 - a-6 - 9a-6z2 + 41a-6z4 - 50a-6z6 + 11a-6z8 + 4a-6z10 + a-5z - 12a-5z3 + 46a-5z5 - 56a-5z7 + 19a-5z9 - 3a-4 - 2a-4z2 + 34a-4z4 - 33a-4z6 + a-4z8 + 4a-4z10 + a-3z - 7a-3z3 + 26a-3z5 - 27a-3z7 + 8a-3z9 - 3a-2 + 2a-2z2 + 12a-2z4 - 16a-2z6 + 5a-2z8 - 2a-1z5 + a-1z7

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

{0, -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=4 is the signature of 11348. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 21                       1 j = 19                     3   j = 17                   6 1   j = 15                 9 3     j = 13               11 6       j = 11             12 9         j = 9           11 11           j = 7         9 12             j = 5       6 11               j = 3     4 10                 j = 1   1 5                   j = -1   4                     j = -3 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, 348]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 348]]
Out[3]=	PD[X[6, 2, 7, 1], X[18, 4, 19, 3], X[16, 5, 17, 6], X[12, 8, 13, 7], > X[4, 10, 5, 9], X[2, 11, 3, 12], X[20, 14, 21, 13], X[22, 16, 1, 15], > X[10, 18, 11, 17], X[8, 19, 9, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 348]]
Out[4]=	GaussCode[1, -6, 2, -5, 3, -1, 4, -10, 5, -9, 6, -4, 7, -11, 8, -3, 9, -2, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 348]]
Out[5]=	DTCode[6, 18, 16, 12, 4, 2, 20, 22, 10, 8, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 348]][t]
Out[6]=	-4 7 19 29 2 3 4 33 + t - -- + -- - -- - 29 t + 19 t - 7 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 348]][z]
Out[7]=	4 6 8 1 - 3 z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 348]}
In[9]:=	{KnotDet[Knot[11, Alternating, 348]], KnotSignature[Knot[11, Alternating, 348]]}
Out[9]=	{145, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 348]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 10 5 - - - 9 q + 15 q - 20 q + 23 q - 23 q + 20 q - 15 q + 9 q - 4 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 348]}
In[12]:=	A2Invariant[Knot[11, Alternating, 348]][q]
Out[12]=	-2 2 4 6 8 10 12 14 16 18 20 3 - q - q + 3 q + 2 q - 3 q + 4 q - 6 q + 2 q - q - q + 4 q - 22 24 26 28 30 > 3 q + 2 q - q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 348]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 6 -6 3 3 2 z 3 z z 2 z z 6 z 4 z 2 z 2 z a - -- + -- + ---- - ---- - -- + ---- + -- - ---- + ---- - ---- - ---- + 4 2 8 6 4 2 8 6 4 2 6 a a a a a a a a a a a 6 6 8 4 z z z > ---- - -- + -- 4 2 4 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 348]][a, z]
Out[14]=	2 2 2 2 2 3 -6 3 3 z z z z 2 z 3 z 9 z 2 z 2 z z -a - -- - -- - -- - -- + -- + -- + ---- - ---- - ---- - ---- + ---- - --- + 4 2 9 7 5 3 10 8 6 4 2 11 a a a a a a a a a a a a 3 3 3 3 4 4 4 4 4 4 9 z 5 z 12 z 7 z z 7 z 11 z 41 z 34 z 12 z > ---- + ---- - ----- - ---- + --- - ---- + ----- + ----- + ----- + ----- + 9 7 5 3 12 10 8 6 4 2 a a a a a a a a a a 5 5 5 5 5 5 6 6 6 6 4 z 19 z 5 z 46 z 26 z 2 z 9 z 24 z 50 z 33 z > ---- - ----- - ---- + ----- + ----- - ---- + ---- - ----- - ----- - ----- - 11 9 7 5 3 a 10 8 6 4 a a a a a a a a a 6 7 7 7 7 7 8 8 8 8 16 z 14 z 14 z 56 z 27 z z 15 z 11 z z 5 z > ----- + ----- - ----- - ----- - ----- + -- + ----- + ----- + -- + ---- + 2 9 7 5 3 a 8 6 4 2 a a a a a a a a a 9 9 9 10 10 11 z 19 z 8 z 4 z 4 z > ----- + ----- + ---- + ----- + ----- 7 5 3 6 4 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 348]], Vassiliev[3][Knot[11, Alternating, 348]]}
Out[15]=	{0, -1}
In[16]:=	Kh[Knot[11, Alternating, 348]][q, t]
Out[16]=	3 3 5 1 4 q 5 q 4 q 5 7 7 2 10 q + 6 q + ----- + ---- + -- + --- + ---- + 11 q t + 9 q t + 12 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 > 11 q t + 11 q t + 12 q t + 9 q t + 11 q t + 6 q t + 15 5 15 6 17 6 17 7 19 7 21 8 > 9 q t + 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a348

K11a347 K11a349