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

K11a315 K11a317

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 16 20 22 4 18 10 2 14 8

Alexander Polynomial:

t-4 - 5t-3 + 14t-2 - 25t-1 + 31 - 25t + 14t2 - 5t3 + t4

Conway Polynomial:

1 + 2z2 + 4z4 + 3z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{K11a35, ...}

Determinant and Signature:

{121, 0}

Jones Polynomial:

- q-5 + 3q-4 - 7q-3 + 12q-2 - 16q-1 + 20 - 19q + 17q2 - 13q3 + 8q4 - 4q5 + q6

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

{K11a35, K11a36, ...}

A2 (sl(3)) Invariant:

- q-14 + q-12 - 3q-10 + q-8 + q-6 - 2q-4 + 6q-2 - 1 + 4q2 - 2q6 + 2q8 - 4q10 + q12 - q16 + q18

HOMFLY-PT Polynomial:

a-4 + 2a-4z2 + a-4z4 - 5a-2 - 11a-2z2 - 8a-2z4 - 2a-2z6 + 8 + 17z2 + 15z4 + 6z6 + z8 - 3a2 - 6a2z2 - 4a2z4 - a2z6

Kauffman Polynomial:

a-6z2 - 2a-6z4 + a-6z6 + 6a-5z3 - 10a-5z5 + 4a-5z7 + a-4 - a-4z2 + 5a-4z4 - 13a-4z6 + 6a-4z8 - a-3z + 3a-3z3 - 5a-3z5 - 6a-3z7 + 5a-3z9 + 5a-2 - 18a-2z2 + 27a-2z4 - 26a-2z6 + 7a-2z8 + 2a-2z10 - 3a-1z - 5a-1z3 + 23a-1z5 - 26a-1z7 + 11a-1z9 + 8 - 30z2 + 52z4 - 36z6 + 9z8 + 2z10 - 5az + 10az3 + 4az5 - 10az7 + 6az9 + 3a2 - 14a2z2 + 27a2z4 - 21a2z6 + 8a2z8 - 3a3z + 10a3z3 - 13a3z5 + 6a3z7 - 5a4z4 + 3a4z6 - 2a5z3 + a5z5

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

{2, -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 11316. 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                     3   j = 9                   5 1   j = 7                 8 3     j = 5               9 5       j = 3             10 8         j = 1           10 9           j = -1         7 11             j = -3       5 9               j = -5     2 7                 j = -7   1 5                   j = -9   2                     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, 316]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 316]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 4, 13, 3], X[16, 5, 17, 6], X[20, 8, 21, 7], > X[22, 10, 1, 9], X[4, 12, 5, 11], X[18, 13, 19, 14], X[10, 15, 11, 16], > X[2, 17, 3, 18], X[14, 19, 15, 20], X[8, 22, 9, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 316]]
Out[4]=	GaussCode[1, -9, 2, -6, 3, -1, 4, -11, 5, -8, 6, -2, 7, -10, 8, -3, 9, -7, 10, > -4, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 316]]
Out[5]=	DTCode[6, 12, 16, 20, 22, 4, 18, 10, 2, 14, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 316]][t]
Out[6]=	-4 5 14 25 2 3 4 31 + t - -- + -- - -- - 25 t + 14 t - 5 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 316]][z]
Out[7]=	2 4 6 8 1 + 2 z + 4 z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 35], Knot[11, Alternating, 316]}
In[9]:=	{KnotDet[Knot[11, Alternating, 316]], KnotSignature[Knot[11, Alternating, 316]]}
Out[9]=	{121, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 316]][q]
Out[10]=	-5 3 7 12 16 2 3 4 5 6 20 - q + -- - -- + -- - -- - 19 q + 17 q - 13 q + 8 q - 4 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, 35], Knot[11, Alternating, 36], > Knot[11, Alternating, 316]}
In[12]:=	A2Invariant[Knot[11, Alternating, 316]][q]
Out[12]=	-14 -12 3 -8 -6 2 6 2 6 8 10 -1 - q + q - --- + q + q - -- + -- + 4 q - 2 q + 2 q - 4 q + 10 4 2 q q q 12 16 18 > q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 316]][a, z]
Out[13]=	2 2 4 4 -4 5 2 2 2 z 11 z 2 2 4 z 8 z 8 + a - -- - 3 a + 17 z + ---- - ----- - 6 a z + 15 z + -- - ---- - 2 4 2 4 2 a a a a a 6 2 4 6 2 z 2 6 8 > 4 a z + 6 z - ---- - a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 316]][a, z]
Out[14]=	2 2 2 -4 5 2 z 3 z 3 2 z z 18 z 8 + a + -- + 3 a - -- - --- - 5 a z - 3 a z - 30 z + -- - -- - ----- - 2 3 a 6 4 2 a a a a a 3 3 3 2 2 6 z 3 z 5 z 3 3 3 5 3 4 > 14 a z + ---- + ---- - ---- + 10 a z + 10 a z - 2 a z + 52 z - 5 3 a a a 4 4 4 5 5 5 2 z 5 z 27 z 2 4 4 4 10 z 5 z 23 z 5 > ---- + ---- + ----- + 27 a z - 5 a z - ----- - ---- + ----- + 4 a z - 6 4 2 5 3 a a a a a a 6 6 6 7 3 5 5 5 6 z 13 z 26 z 2 6 4 6 4 z > 13 a z + a z - 36 z + -- - ----- - ----- - 21 a z + 3 a z + ---- - 6 4 2 5 a a a a 7 7 8 8 9 6 z 26 z 7 3 7 8 6 z 7 z 2 8 5 z > ---- - ----- - 10 a z + 6 a z + 9 z + ---- + ---- + 8 a z + ---- + 3 a 4 2 3 a a a a 9 10 11 z 9 10 2 z > ----- + 6 a z + 2 z + ----- a 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 316]], Vassiliev[3][Knot[11, Alternating, 316]]}
Out[15]=	{2, -1}
In[16]:=	Kh[Knot[11, Alternating, 316]][q, t]
Out[16]=	11 1 2 1 5 2 7 5 9 -- + 10 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 7 3 3 2 5 2 5 3 7 3 7 4 > --- + 9 q t + 10 q t + 8 q t + 9 q t + 5 q t + 8 q t + 3 q t + q t 9 4 9 5 11 5 13 6 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a316

K11a315 K11a317