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

K11n171 K11n173

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,3,11,4 X5,16,6,17 X12,8,13,7 X20,10,21,9 X2,11,3,12 X18,13,19,14 X15,4,16,5 X22,17,1,18 X8,20,9,19 X14,21,15,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 -16 12 20 2 18 -4 22 8 14

Alexander Polynomial:

- t-3 + 5t-2 - 11t-1 + 15 - 11t + 5t2 - t3

Conway Polynomial:

1 - z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{927, K11n4, K11n21, ...}

Determinant and Signature:

{49, 0}

Jones Polynomial:

- q-7 + 3q-6 - 5q-5 + 7q-4 - 8q-3 + 8q-2 - 7q-1 + 6 - 3q + q2

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

{...}

A2 (sl(3)) Invariant:

- q-22 + q-18 - q-16 + 2q-14 + q-8 - 2q-6 + q-4 - q-2 + 1 + 2q2 - q4 + q6

HOMFLY-PT Polynomial:

2 + 2z2 + z4 - 3a2 - 6a2z2 - 4a2z4 - a2z6 + 3a4 + 5a4z2 + 2a4z4 - a6 - a6z2

Kauffman Polynomial:

a-2z2 - a-1z + 3a-1z3 + 2 - 3z2 + 2z4 + z6 - 3az + 10az3 - 11az5 + 4az7 + 3a2 - 14a2z2 + 22a2z4 - 18a2z6 + 5a2z8 - 5a3z + 17a3z3 - 15a3z5 - a3z7 + 2a3z9 + 3a4 - 16a4z2 + 36a4z4 - 32a4z6 + 8a4z8 - 5a5z + 15a5z3 - 8a5z5 - 4a5z7 + 2a5z9 + a6 - 6a6z2 + 16a6z4 - 13a6z6 + 3a6z8 - 2a7z + 5a7z3 - 4a7z5 + a7z7

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=0 is the signature of 11172. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 5                   1 j = 3                 2   j = 1               4 1   j = -1             4 3     j = -3           4 3       j = -5         4 4         j = -7       3 4           j = -9     2 4             j = -11   1 3               j = -13   2                 j = -15 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, NonAlternating, 172]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 172]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[5, 16, 6, 17], X[12, 8, 13, 7], > X[20, 10, 21, 9], X[2, 11, 3, 12], X[18, 13, 19, 14], X[15, 4, 16, 5], > X[22, 17, 1, 18], X[8, 20, 9, 19], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 172]]
Out[4]=	GaussCode[1, -6, 2, 8, -3, -1, 4, -10, 5, -2, 6, -4, 7, -11, -8, 3, 9, -7, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 172]]
Out[5]=	DTCode[6, 10, -16, 12, 20, 2, 18, -4, 22, 8, 14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 172]][t]
Out[6]=	-3 5 11 2 3 15 - t + -- - -- - 11 t + 5 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 172]][z]
Out[7]=	4 6 1 - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 27], Knot[11, NonAlternating, 4], Knot[11, NonAlternating, 21], > Knot[11, NonAlternating, 172]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 172]], KnotSignature[Knot[11, NonAlternating, 172]]}
Out[9]=	{49, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 172]][q]
Out[10]=	-7 3 5 7 8 8 7 2 6 - q + -- - -- + -- - -- + -- - - - 3 q + q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 172]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 172]][q]
Out[12]=	-22 -18 -16 2 -8 2 -4 -2 2 4 6 1 - q + q - q + --- + q - -- + q - q + 2 q - q + q 14 6 q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 172]][a, z]
Out[13]=	2 4 6 2 2 2 4 2 6 2 4 2 4 2 - 3 a + 3 a - a + 2 z - 6 a z + 5 a z - a z + z - 4 a z + 4 4 2 6 > 2 a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 172]][a, z]
Out[14]=	2 2 4 6 z 3 5 7 2 z 2 + 3 a + 3 a + a - - - 3 a z - 5 a z - 5 a z - 2 a z - 3 z + -- - a 2 a 3 2 2 4 2 6 2 3 z 3 3 3 5 3 > 14 a z - 16 a z - 6 a z + ---- + 10 a z + 17 a z + 15 a z + a 7 3 4 2 4 4 4 6 4 5 3 5 > 5 a z + 2 z + 22 a z + 36 a z + 16 a z - 11 a z - 15 a z - 5 5 7 5 6 2 6 4 6 6 6 7 3 7 > 8 a z - 4 a z + z - 18 a z - 32 a z - 13 a z + 4 a z - a z - 5 7 7 7 2 8 4 8 6 8 3 9 5 9 > 4 a z + a z + 5 a z + 8 a z + 3 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 172]], Vassiliev[3][Knot[11, NonAlternating, 172]]}
Out[15]=	{0, -1}
In[16]:=	Kh[Knot[11, NonAlternating, 172]][q, t]
Out[16]=	3 1 2 1 3 2 4 3 4 - + 4 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 4 4 4 3 4 3 5 2 > ----- + ----- + ----- + ---- + --- + q t + 2 q t + q t 5 3 5 2 3 2 3 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n172

K11n171 K11n173