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

K11n9 K11n11

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X7,14,8,15 X2,9,3,10 X18,11,19,12 X13,6,14,7 X20,15,21,16 X22,17,1,18 X12,19,13,20 X16,21,17,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 7t-2 - 15t-1 + 19 - 15t + 7t2 - t3

Conway Polynomial:

1 + 4z2 + z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{K11n103, K11n144, ...}

Determinant and Signature:

{65, -4}

Jones Polynomial:

- q-11 + 3q-10 - 6q-9 + 9q-8 - 11q-7 + 11q-6 - 10q-5 + 8q-4 - 4q-3 + 2q-2

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

{K11n144, ...}

A2 (sl(3)) Invariant:

- q-34 + q-30 - 2q-28 + 2q-26 - q-22 + q-20 - 3q-18 + 2q-16 - q-14 + q-12 + 3q-10 - q-8 + 2q-6

HOMFLY-PT Polynomial:

3a4 + 5a4z2 + 2a4z4 - 3a6 - 4a6z2 - 3a6z4 - a6z6 + 2a8 + 4a8z2 + 2a8z4 - a10 - a10z2

Kauffman Polynomial:

3a4 - 6a4z2 + 3a4z4 - a5z - a5z3 + a5z5 + a5z7 + 3a6 - 9a6z2 + 8a6z4 - 3a6z6 + 2a6z8 - a7z3 - a7z5 + 2a7z7 + a7z9 + 2a8 - 6a8z2 + 7a8z4 - 7a8z6 + 5a8z8 + 3a9z3 - 9a9z5 + 5a9z7 + a9z9 + a10 - 4a10z4 - a10z6 + 3a10z8 + a11z3 - 6a11z5 + 4a11z7 + 3a12z2 - 6a12z4 + 3a12z6 + a13z - 2a13z3 + a13z5

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

{4, -9}

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 1110. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3                   2 j = -5                 3 1 j = -7               5 1   j = -9             5 3     j = -11           6 5       j = -13         5 5         j = -15       4 6           j = -17     2 5             j = -19   1 4               j = -21   2                 j = -23 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, 10]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 10]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[7, 14, 8, 15], > X[2, 9, 3, 10], X[18, 11, 19, 12], X[13, 6, 14, 7], X[20, 15, 21, 16], > X[22, 17, 1, 18], X[12, 19, 13, 20], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 10]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, 7, -4, -2, 5, -3, 6, -10, -7, 4, 8, -11, 9, -6, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 10]]
Out[5]=	DTCode[4, 8, 10, -14, 2, 18, -6, 20, 22, 12, 16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 10]][t]
Out[6]=	-3 7 15 2 3 19 - t + -- - -- - 15 t + 7 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 10]][z]
Out[7]=	2 4 6 1 + 4 z + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 10], Knot[11, NonAlternating, 103], > Knot[11, NonAlternating, 144]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 10]], KnotSignature[Knot[11, NonAlternating, 10]]}
Out[9]=	{65, -4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 10]][q]
Out[10]=	-11 3 6 9 11 11 10 8 4 2 -q + --- - -- + -- - -- + -- - -- + -- - -- + -- 10 9 8 7 6 5 4 3 2 q q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 10], Knot[11, NonAlternating, 144]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 10]][q]
Out[12]=	-34 -30 2 2 -22 -20 3 2 -14 -12 3 -8 -q + q - --- + --- - q + q - --- + --- - q + q + --- - q + 28 26 18 16 10 q q q q q 2 > -- 6 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 10]][a, z]
Out[13]=	4 6 8 10 4 2 6 2 8 2 10 2 4 4 3 a - 3 a + 2 a - a + 5 a z - 4 a z + 4 a z - a z + 2 a z - 6 4 8 4 6 6 > 3 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 10]][a, z]
Out[14]=	4 6 8 10 5 13 4 2 6 2 8 2 3 a + 3 a + 2 a + a - a z + a z - 6 a z - 9 a z - 6 a z + 12 2 5 3 7 3 9 3 11 3 13 3 4 4 > 3 a z - a z - a z + 3 a z + a z - 2 a z + 3 a z + 6 4 8 4 10 4 12 4 5 5 7 5 9 5 > 8 a z + 7 a z - 4 a z - 6 a z + a z - a z - 9 a z - 11 5 13 5 6 6 8 6 10 6 12 6 5 7 > 6 a z + a z - 3 a z - 7 a z - a z + 3 a z + a z + 7 7 9 7 11 7 6 8 8 8 10 8 7 9 9 9 > 2 a z + 5 a z + 4 a z + 2 a z + 5 a z + 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 10]], Vassiliev[3][Knot[11, NonAlternating, 10]]}
Out[15]=	{4, -9}
In[16]:=	Kh[Knot[11, NonAlternating, 10]][q, t]
Out[16]=	-5 2 1 2 1 4 2 5 4 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q q t q t q t q t q t q t q t 6 5 5 6 5 5 3 5 1 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t q t 3 > ---- 5 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n10

K11n9 K11n11