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

K11a313 K11a315

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 16 18 14 4 20 22 2 8 10

Alexander Polynomial:

- t-4 + 7t-3 - 21t-2 + 36t-1 - 41 + 36t - 21t2 + 7t3 - t4

Conway Polynomial:

1 - z2 + z4 - z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{171, 2}

Jones Polynomial:

q-3 - 5q-2 + 12q-1 - 18 + 25q - 28q2 + 27q3 - 24q4 + 17q5 - 9q6 + 4q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-8 - 3q-6 + 4q-4 - q-2 + 1 + 5q2 - 6q4 + 5q6 - 5q8 + q10 + q12 - 4q14 + 5q16 - 2q18 + q20 + q22 - q24

HOMFLY-PT Polynomial:

- 2a-6z2 - a-6z4 + a-4 + 5a-4z2 + 6a-4z4 + 2a-4z6 - 2a-2 - 5a-2z2 - 6a-2z4 - 4a-2z6 - a-2z8 + 2 + z2 + 2z4 + z6

Kauffman Polynomial:

- a-9z3 + a-9z5 + 2a-8z2 - 5a-8z4 + 4a-8z6 - a-7z + 5a-7z3 - 10a-7z5 + 8a-7z7 + a-6z2 + 6a-6z4 - 14a-6z6 + 11a-6z8 - 4a-5z + 8a-5z3 - 3a-5z5 - 9a-5z7 + 10a-5z9 + a-4 - 8a-4z2 + 34a-4z4 - 44a-4z6 + 15a-4z8 + 4a-4z10 - 4a-3z + 21a-3z5 - 41a-3z7 + 21a-3z9 + 2a-2 - 10a-2z2 + 38a-2z4 - 51a-2z6 + 15a-2z8 + 4a-2z10 - a-1z + 5a-1z5 - 19a-1z7 + 11a-1z9 + 2 - 3z2 + 14z4 - 24z6 + 11z8 + 2az3 - 8az5 + 5az7 - a2z4 + a2z6

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=2 is the signature of 11314. 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 = 17                       1 j = 15                     3   j = 13                   6 1   j = 11                 11 3     j = 9               13 6       j = 7             14 11         j = 5           14 13           j = 3         11 14             j = 1       8 15               j = -1     4 10                 j = -3   1 8                   j = -5   4                     j = -7 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, 314]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 314]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 4, 13, 3], X[16, 5, 17, 6], X[18, 8, 19, 7], > X[14, 10, 15, 9], X[4, 12, 5, 11], X[20, 13, 21, 14], X[22, 16, 1, 15], > X[2, 17, 3, 18], X[8, 20, 9, 19], X[10, 21, 11, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 314]]
Out[4]=	GaussCode[1, -9, 2, -6, 3, -1, 4, -10, 5, -11, 6, -2, 7, -5, 8, -3, 9, -4, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 314]]
Out[5]=	DTCode[6, 12, 16, 18, 14, 4, 20, 22, 2, 8, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 314]][t]
Out[6]=	-4 7 21 36 2 3 4 -41 - t + -- - -- + -- + 36 t - 21 t + 7 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 314]][z]
Out[7]=	2 4 6 8 1 - z + z - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 314]}
In[9]:=	{KnotDet[Knot[11, Alternating, 314]], KnotSignature[Knot[11, Alternating, 314]]}
Out[9]=	{171, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 314]][q]
Out[10]=	-3 5 12 2 3 4 5 6 7 8 -18 + q - -- + -- + 25 q - 28 q + 27 q - 24 q + 17 q - 9 q + 4 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 314]}
In[12]:=	A2Invariant[Knot[11, Alternating, 314]][q]
Out[12]=	-8 3 4 -2 2 4 6 8 10 12 14 1 + q - -- + -- - q + 5 q - 6 q + 5 q - 5 q + q + q - 4 q + 6 4 q q 16 18 20 22 24 > 5 q - 2 q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 314]][a, z]
Out[13]=	2 2 2 4 4 4 6 -4 2 2 2 z 5 z 5 z 4 z 6 z 6 z 6 2 z 2 + a - -- + z - ---- + ---- - ---- + 2 z - -- + ---- - ---- + z + ---- - 2 6 4 2 6 4 2 4 a a a a a a a a 6 8 4 z z > ---- - -- 2 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 314]][a, z]
Out[14]=	2 2 2 2 3 -4 2 z 4 z 4 z z 2 2 z z 8 z 10 z z 2 + a + -- - -- - --- - --- - - - 3 z + ---- + -- - ---- - ----- - -- + 2 7 5 3 a 8 6 4 2 9 a a a a a a a a a 3 3 4 4 4 4 5 5 z 8 z 3 4 5 z 6 z 34 z 38 z 2 4 z > ---- + ---- + 2 a z + 14 z - ---- + ---- + ----- + ----- - a z + -- - 7 5 8 6 4 2 9 a a a a a a a 5 5 5 5 6 6 6 10 z 3 z 21 z 5 z 5 6 4 z 14 z 44 z > ----- - ---- + ----- + ---- - 8 a z - 24 z + ---- - ----- - ----- - 7 5 3 a 8 6 4 a a a a a a 6 7 7 7 7 8 51 z 2 6 8 z 9 z 41 z 19 z 7 8 11 z > ----- + a z + ---- - ---- - ----- - ----- + 5 a z + 11 z + ----- + 2 7 5 3 a 6 a a a a a 8 8 9 9 9 10 10 15 z 15 z 10 z 21 z 11 z 4 z 4 z > ----- + ----- + ----- + ----- + ----- + ----- + ----- 4 2 5 3 a 4 2 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 314]], Vassiliev[3][Knot[11, Alternating, 314]]}
Out[15]=	{-1, -1}
In[16]:=	Kh[Knot[11, Alternating, 314]][q, t]
Out[16]=	3 1 4 1 8 4 10 8 q 3 15 q + 11 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 14 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 14 q t + 13 q t + 14 q t + 11 q t + 13 q t + 6 q t + 11 q t + 11 5 13 5 13 6 15 6 17 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 K11a314

K11a313 K11a315