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

K11a70 K11a72

Knotscape

This page is passe. Go here instead!    The Knot K11a71 Visit K11a71's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X12,5,13,6 X14,8,15,7 X2,10,3,9 X22,11,1,12 X18,13,19,14 X20,16,21,15 X6,18,7,17 X8,19,9,20 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 14 2 22 18 20 6 8 16

Alexander Polynomial:

- t-4 + 6t-3 - 18t-2 + 34t-1 - 41 + 34t - 18t2 + 6t3 - t4

Conway Polynomial:

1 - 2z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a248, ...}

Determinant and Signature:

{159, 2}

Jones Polynomial:

q-3 - 4q-2 + 9q-1 - 15 + 22q - 25q2 + 26q3 - 23q4 + 17q5 - 11q6 + 5q7 - q8

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

{K11a248, ...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- a-6z2 - a-6z4 + 5a-4z2 + 6a-4z4 + 2a-4z6 - 7a-2z2 - 10a-2z4 - 5a-2z6 - a-2z8 + 1 + 3z2 + 3z4 + z6

Kauffman Polynomial:

a-9z5 - 4a-8z4 + 5a-8z6 + 5a-7z3 - 15a-7z5 + 11a-7z7 - a-6z2 + 5a-6z4 - 17a-6z6 + 13a-6z8 - a-5z + 17a-5z3 - 29a-5z5 + 5a-5z7 + 8a-5z9 - 8a-4z2 + 35a-4z4 - 53a-4z6 + 23a-4z8 + 2a-4z10 - 3a-3z + 20a-3z3 - 18a-3z5 - 13a-3z7 + 14a-3z9 - 12a-2z2 + 39a-2z4 - 47a-2z6 + 17a-2z8 + 2a-2z10 - 3a-1z + 14a-1z3 - 14a-1z5 - 3a-1z7 + 6a-1z9 + 1 - 4z2 + 11z4 - 15z6 + 7z8 - az + 6az3 - 9az5 + 4az7 + a2z2 - 2a2z4 + a2z6

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

{0, 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=2 is the signature of 1171. 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                     4   j = 13                   7 1   j = 11                 10 4     j = 9               13 7       j = 7             13 10         j = 5           12 13           j = 3         10 13             j = 1       6 13               j = -1     3 9                 j = -3   1 6                   j = -5   3                     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, 71]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 71]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[12, 5, 13, 6], X[14, 8, 15, 7], > X[2, 10, 3, 9], X[22, 11, 1, 12], X[18, 13, 19, 14], X[20, 16, 21, 15], > X[6, 18, 7, 17], X[8, 19, 9, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 71]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -10, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, > -8, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 71]]
Out[5]=	DTCode[4, 10, 12, 14, 2, 22, 18, 20, 6, 8, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 71]][t]
Out[6]=	-4 6 18 34 2 3 4 -41 - t + -- - -- + -- + 34 t - 18 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 71]][z]
Out[7]=	4 6 8 1 - 2 z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 71], Knot[11, Alternating, 248]}
In[9]:=	{KnotDet[Knot[11, Alternating, 71]], KnotSignature[Knot[11, Alternating, 71]]}
Out[9]=	{159, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 71]][q]
Out[10]=	-3 4 9 2 3 4 5 6 7 8 -15 + q - -- + - + 22 q - 25 q + 26 q - 23 q + 17 q - 11 q + 5 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 71], Knot[11, Alternating, 248]}
In[12]:=	A2Invariant[Knot[11, Alternating, 71]][q]
Out[12]=	-8 2 3 2 2 4 6 8 12 14 16 -1 + q - -- + -- - -- + 5 q - 4 q + 6 q - 2 q + q - 5 q + 4 q - 6 4 2 q q q 18 22 24 > 2 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 71]][a, z]
Out[13]=	2 2 2 4 4 4 6 6 8 2 z 5 z 7 z 4 z 6 z 10 z 6 2 z 5 z z 1 + 3 z - -- + ---- - ---- + 3 z - -- + ---- - ----- + z + ---- - ---- - -- 6 4 2 6 4 2 4 2 2 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 71]][a, z]
Out[14]=	2 2 2 3 3 z 3 z 3 z 2 z 8 z 12 z 2 2 5 z 17 z 1 - -- - --- - --- - a z - 4 z - -- - ---- - ----- + a z + ---- + ----- + 5 3 a 6 4 2 7 5 a a a a a a a 3 3 4 4 4 4 20 z 14 z 3 4 4 z 5 z 35 z 39 z 2 4 > ----- + ----- + 6 a z + 11 z - ---- + ---- + ----- + ----- - 2 a z + 3 a 8 6 4 2 a a a a a 5 5 5 5 5 6 6 z 15 z 29 z 18 z 14 z 5 6 5 z 17 z > -- - ----- - ----- - ----- - ----- - 9 a z - 15 z + ---- - ----- - 9 7 5 3 a 8 6 a a a a a a 6 6 7 7 7 7 53 z 47 z 2 6 11 z 5 z 13 z 3 z 7 8 > ----- - ----- + a z + ----- + ---- - ----- - ---- + 4 a z + 7 z + 4 2 7 5 3 a a a a a a 8 8 8 9 9 9 10 10 13 z 23 z 17 z 8 z 14 z 6 z 2 z 2 z > ----- + ----- + ----- + ---- + ----- + ---- + ----- + ----- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 71]], Vassiliev[3][Knot[11, Alternating, 71]]}
Out[15]=	{0, 0}
In[16]:=	Kh[Knot[11, Alternating, 71]][q, t]
Out[16]=	3 1 3 1 6 3 9 6 q 3 13 q + 10 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 13 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 > 12 q t + 13 q t + 13 q t + 10 q t + 13 q t + 7 q t + 10 q t + 11 5 13 5 13 6 15 6 17 7 > 4 q t + 7 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a71

K11a70 K11a72