The Alternating Knot 10₇₂

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Acknowledgement

KnotPlot

PD Presentation: X₄₂₅₁ X_10,6,11,5 X₈₃₉₄ X_2,9,3,10 X_16,8,17,7 X_18,12,19,11 X_20,14,1,13 X_12,20,13,19 X_14,18,15,17 X_6,16,7,15

Gauss Code: {1, -4, 3, -1, 2, -10, 5, -3, 4, -2, 6, -8, 7, -9, 10, -5, 9, -6, 8, -7}

DT (Dowker-Thistlethwaite) Code: 4 8 10 16 2 18 20 6 14 12

Minimum Braid Representative:

Length is 11, width is 4
Braid index is 4

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index

Reversible 2 3 3 / NotAvailable 1

Alexander Polynomial: - 2t^-3 + 9t^-2 - 16t^-1 + 19 - 16t + 9t² - 2t³

Conway Polynomial: 1 + 2z² - 3z⁴ - 2z⁶

Other knots with the same Alexander/Conway Polynomial: {...}

Determinant and Signature: {73, 4}

Jones Polynomial: 1 - 2q + 5q² - 8q³ + 11q⁴ - 12q⁵ + 12q⁶ - 10q⁷ + 7q⁸ - 4q⁹ + q¹⁰

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

A2 (sl(3)) Invariant: 1 + q⁴ + 2q⁶ - 2q⁸ + 2q¹⁰ - 2q¹² + 2q¹⁶ - q¹⁸ + 3q²⁰ - 2q²² - 2q²⁸ + q³⁰

HOMFLY-PT Polynomial: - a^-8 + a^-8z² + a^-8z⁴ + 2a^-6 + a^-6z² - 2a^-6z⁴ - a^-6z⁶ - 2a^-4 - 3a^-4z² - 3a^-4z⁴ - a^-4z⁶ + 2a^-2 + 3a^-2z² + a^-2z⁴

Kauffman Polynomial: a^-12z⁴ - 3a^-11z³ + 4a^-11z⁵ + 2a^-10z² - 8a^-10z⁴ + 7a^-10z⁶ + a^-9z - 7a^-9z⁵ + 7a^-9z⁷ - a^-8 + 6a^-8z² - 11a^-8z⁴ + 2a^-8z⁶ + 4a^-8z⁸ - a^-7z + 9a^-7z³ - 19a^-7z⁵ + 9a^-7z⁷ + a^-7z⁹ - 2a^-6 + 7a^-6z² - 4a^-6z⁴ - 8a^-6z⁶ + 6a^-6z⁸ - 3a^-5z + 11a^-5z³ - 14a^-5z⁵ + 4a^-5z⁷ + a^-5z⁹ - 2a^-4 + 8a^-4z² - 6a^-4z⁴ - 2a^-4z⁶ + 2a^-4z⁸ - a^-3z + 5a^-3z³ - 6a^-3z⁵ + 2a^-3z⁷ - 2a^-2 + 5a^-2z² - 4a^-2z⁴ + a^-2z⁶

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {2, 4}

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

t^rq^j r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8

j = 21 1

j = 19 3

j = 17 4 1

j = 15 6 3

j = 13 6 4

j = 11 6 6

j = 9 5 6

j = 7 3 6

j = 5 2 5

j = 3 1 4

j = 1 1

j = -1 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-2 - 2q^-1 + 7q - 10q² - 4q³ + 28q⁴ - 25q⁵ - 22q⁶ + 67q⁷ - 36q⁸ - 59q⁹ + 108q¹⁰ - 32q¹¹ - 97q¹² + 128q¹³ - 16q¹⁴ - 115q¹⁵ + 117q¹⁶ + 2q¹⁷ - 101q¹⁸ + 80q¹⁹ + 12q²⁰ - 63q²¹ + 37q²² + 10q²³ - 24q²⁴ + 10q²⁵ + 3q²⁶ - 4q²⁷ + q²⁸

3 q^-6 - 2q^-5 + 2q^-3 + 4q^-2 - 9q^-1 - 5 + 13q + 19q² - 28q³ - 34q⁴ + 34q⁵ + 76q⁶ - 48q⁷ - 120q⁸ + 33q⁹ + 196q¹⁰ - 11q¹¹ - 266q¹² - 48q¹³ + 344q¹⁴ + 123q¹⁵ - 399q¹⁶ - 227q¹⁷ + 446q¹⁸ + 325q¹⁹ - 456q²⁰ - 434q²¹ + 454q²² + 521q²³ - 424q²⁴ - 596q²⁵ + 385q²⁶ + 636q²⁷ - 321q²⁸ - 656q²⁹ + 256q³⁰ + 633q³¹ - 179q³² - 583q³³ + 108q³⁴ + 502q³⁵ - 46q³⁶ - 401q³⁷ - 2q³⁸ + 302q³⁹ + 22q⁴⁰ - 202q⁴¹ - 33q⁴² + 128q⁴³ + 25q⁴⁴ - 69q⁴⁵ - 21q⁴⁶ + 40q⁴⁷ + 7q⁴⁸ - 16q⁴⁹ - 4q⁵⁰ + 6q⁵¹ + 3q⁵² - 4q⁵³ + q⁵⁴

4 q^-12 - 2q^-11 + 2q^-9 - q^-8 + 5q^-7 - 11q^-6 - q^-5 + 13q^-4 - q^-3 + 18q^-2 - 44q^-1 - 19 + 43q + 25q² + 73q³ - 127q⁴ - 112q⁵ + 57q⁶ + 112q⁷ + 285q⁸ - 214q⁹ - 358q¹⁰ - 104q¹¹ + 182q¹² + 791q¹³ - 83q¹⁴ - 667q¹⁵ - 624q¹⁶ - 61q¹⁷ + 1496q¹⁸ + 495q¹⁹ - 686q²⁰ - 1400q²¹ - 879q²² + 1994q²³ + 1420q²⁴ - 113q²⁵ - 2035q²⁶ - 2149q²⁷ + 1960q²⁸ + 2299q²⁹ + 946q³⁰ - 2228q³¹ - 3449q³² + 1433q³³ + 2837q³⁴ + 2126q³⁵ - 2005q³⁶ - 4435q³⁷ + 673q³⁸ + 2985q³⁹ + 3113q⁴⁰ - 1515q⁴¹ - 4939q⁴² - 135q⁴³ + 2753q⁴⁴ + 3734q⁴⁵ - 833q⁴⁶ - 4849q⁴⁷ - 861q⁴⁸ + 2111q⁴⁹ + 3815q⁵⁰ - 41q⁵¹ - 4078q⁵² - 1309q⁵³ + 1154q⁵⁴ + 3244q⁵⁵ + 602q⁵⁶ - 2792q⁵⁷ - 1271q⁵⁸ + 243q⁵⁹ + 2180q⁶⁰ + 812q⁶¹ - 1476q⁶² - 825q⁶³ - 242q⁶⁴ + 1106q⁶⁵ + 601q⁶⁶ - 594q⁶⁷ - 329q⁶⁸ - 277q⁶⁹ + 415q⁷⁰ + 283q⁷¹ - 198q⁷² - 63q⁷³ - 141q⁷⁴ + 123q⁷⁵ + 88q⁷⁶ - 66q⁷⁷ + 5q⁷⁸ - 43q⁷⁹ + 30q⁸⁰ + 20q⁸¹ - 19q⁸² + 4q⁸³ - 8q⁸⁴ + 6q⁸⁵ + 3q⁸⁶ - 4q⁸⁷ + q⁸⁸

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]:=
PD[Knot[10, 72]]

Out[2]=
PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], > X[16, 8, 17, 7], X[18, 12, 19, 11], X[20, 14, 1, 13], X[12, 20, 13, 19], > X[14, 18, 15, 17], X[6, 16, 7, 15]]

In[3]:=
GaussCode[Knot[10, 72]]

Out[3]=
GaussCode[1, -4, 3, -1, 2, -10, 5, -3, 4, -2, 6, -8, 7, -9, 10, -5, 9, -6, 8, > -7]

In[4]:=
DTCode[Knot[10, 72]]

Out[4]=
DTCode[4, 8, 10, 16, 2, 18, 20, 6, 14, 12]

In[5]:=
br = BR[Knot[10, 72]]

Out[5]=
BR[4, {1, 1, 1, 1, 2, 2, -1, 2, -3, 2, -3}]

In[6]:=
{First[br], Crossings[br]}

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 72]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 72]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 72]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Reversible, 2, 3, 3, NotAvailable, 1}

In[10]:=
alex = Alexander[Knot[10, 72]][t]

Out[10]=
2 9 16 2 3 19 - -- + -- - -- - 16 t + 9 t - 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 72]][z]

Out[11]=
2 4 6 1 + 2 z - 3 z - 2 z

In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]

Out[12]=
{Knot[10, 72]}

In[13]:=
{KnotDet[Knot[10, 72]], KnotSignature[Knot[10, 72]]}

Out[13]=
{73, 4}

In[14]:=
Jones[Knot[10, 72]][q]

Out[14]=
2 3 4 5 6 7 8 9 10 1 - 2 q + 5 q - 8 q + 11 q - 12 q + 12 q - 10 q + 7 q - 4 q + q

In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]

Out[15]=
{Knot[10, 72]}

In[16]:=
A2Invariant[Knot[10, 72]][q]

Out[16]=
4 6 8 10 12 16 18 20 22 28 30 1 + q + 2 q - 2 q + 2 q - 2 q + 2 q - q + 3 q - 2 q - 2 q + q

In[17]:=
HOMFLYPT[Knot[10, 72]][a, z]

Out[17]=
2 2 2 2 4 4 4 4 6 6 -8 2 2 2 z z 3 z 3 z z 2 z 3 z z z z -a + -- - -- + -- + -- + -- - ---- + ---- + -- - ---- - ---- + -- - -- - -- 6 4 2 8 6 4 2 8 6 4 2 6 4 a a a a a a a a a a a a a

In[18]:=
Kauffman[Knot[10, 72]][a, z]

Out[18]=
2 2 2 2 2 -8 2 2 2 z z 3 z z 2 z 6 z 7 z 8 z 5 z -a - -- - -- - -- + -- - -- - --- - -- + ---- + ---- + ---- + ---- + ---- - 6 4 2 9 7 5 3 10 8 6 4 2 a a a a a a a a a a a a 3 3 3 3 4 4 4 4 4 4 3 z 9 z 11 z 5 z z 8 z 11 z 4 z 6 z 4 z > ---- + ---- + ----- + ---- + --- - ---- - ----- - ---- - ---- - ---- + 11 7 5 3 12 10 8 6 4 2 a a a a a a a a a a 5 5 5 5 5 6 6 6 6 6 4 z 7 z 19 z 14 z 6 z 7 z 2 z 8 z 2 z z > ---- - ---- - ----- - ----- - ---- + ---- + ---- - ---- - ---- + -- + 11 9 7 5 3 10 8 6 4 2 a a a a a a a a a a 7 7 7 7 8 8 8 9 9 7 z 9 z 4 z 2 z 4 z 6 z 2 z z z > ---- + ---- + ---- + ---- + ---- + ---- + ---- + -- + -- 9 7 5 3 8 6 4 7 5 a a a a a a a a a

In[19]:=
{Vassiliev[2][Knot[10, 72]], Vassiliev[3][Knot[10, 72]]}

Out[19]=
{2, 4}

In[20]:=
Kh[Knot[10, 72]][q, t]

Out[20]=
3 3 5 1 q q 5 7 7 2 9 2 9 3 4 q + 2 q + ---- + - + -- + 5 q t + 3 q t + 6 q t + 5 q t + 6 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 6 q t + 6 q t + 6 q t + 4 q t + 6 q t + 3 q t + 17 6 17 7 19 7 21 8 > 4 q t + q t + 3 q t + q t

In[21]:=
ColouredJones[Knot[10, 72], 2][q]

Out[21]=
-2 2 2 3 4 5 6 7 8 9 q - - + 7 q - 10 q - 4 q + 28 q - 25 q - 22 q + 67 q - 36 q - 59 q + q 10 11 12 13 14 15 16 17 > 108 q - 32 q - 97 q + 128 q - 16 q - 115 q + 117 q + 2 q - 18 19 20 21 22 23 24 25 > 101 q + 80 q + 12 q - 63 q + 37 q + 10 q - 24 q + 10 q + 26 27 28 > 3 q - 4 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₇₂

10₇₁
10₇₃

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-2 - 2q^-1 + 7q - 10q² - 4q³ + 28q⁴ - 25q⁵ - 22q⁶ + 67q⁷ - 36q⁸ - 59q⁹ + 108q¹⁰ - 32q¹¹ - 97q¹² + 128q¹³ - 16q¹⁴ - 115q¹⁵ + 117q¹⁶ + 2q¹⁷ - 101q¹⁸ + 80q¹⁹ + 12q²⁰ - 63q²¹ + 37q²² + 10q²³ - 24q²⁴ + 10q²⁵ + 3q²⁶ - 4q²⁷ + q²⁸
3	q^-6 - 2q^-5 + 2q^-3 + 4q^-2 - 9q^-1 - 5 + 13q + 19q² - 28q³ - 34q⁴ + 34q⁵ + 76q⁶ - 48q⁷ - 120q⁸ + 33q⁹ + 196q¹⁰ - 11q¹¹ - 266q¹² - 48q¹³ + 344q¹⁴ + 123q¹⁵ - 399q¹⁶ - 227q¹⁷ + 446q¹⁸ + 325q¹⁹ - 456q²⁰ - 434q²¹ + 454q²² + 521q²³ - 424q²⁴ - 596q²⁵ + 385q²⁶ + 636q²⁷ - 321q²⁸ - 656q²⁹ + 256q³⁰ + 633q³¹ - 179q³² - 583q³³ + 108q³⁴ + 502q³⁵ - 46q³⁶ - 401q³⁷ - 2q³⁸ + 302q³⁹ + 22q⁴⁰ - 202q⁴¹ - 33q⁴² + 128q⁴³ + 25q⁴⁴ - 69q⁴⁵ - 21q⁴⁶ + 40q⁴⁷ + 7q⁴⁸ - 16q⁴⁹ - 4q⁵⁰ + 6q⁵¹ + 3q⁵² - 4q⁵³ + q⁵⁴
4	q^-12 - 2q^-11 + 2q^-9 - q^-8 + 5q^-7 - 11q^-6 - q^-5 + 13q^-4 - q^-3 + 18q^-2 - 44q^-1 - 19 + 43q + 25q² + 73q³ - 127q⁴ - 112q⁵ + 57q⁶ + 112q⁷ + 285q⁸ - 214q⁹ - 358q¹⁰ - 104q¹¹ + 182q¹² + 791q¹³ - 83q¹⁴ - 667q¹⁵ - 624q¹⁶ - 61q¹⁷ + 1496q¹⁸ + 495q¹⁹ - 686q²⁰ - 1400q²¹ - 879q²² + 1994q²³ + 1420q²⁴ - 113q²⁵ - 2035q²⁶ - 2149q²⁷ + 1960q²⁸ + 2299q²⁹ + 946q³⁰ - 2228q³¹ - 3449q³² + 1433q³³ + 2837q³⁴ + 2126q³⁵ - 2005q³⁶ - 4435q³⁷ + 673q³⁸ + 2985q³⁹ + 3113q⁴⁰ - 1515q⁴¹ - 4939q⁴² - 135q⁴³ + 2753q⁴⁴ + 3734q⁴⁵ - 833q⁴⁶ - 4849q⁴⁷ - 861q⁴⁸ + 2111q⁴⁹ + 3815q⁵⁰ - 41q⁵¹ - 4078q⁵² - 1309q⁵³ + 1154q⁵⁴ + 3244q⁵⁵ + 602q⁵⁶ - 2792q⁵⁷ - 1271q⁵⁸ + 243q⁵⁹ + 2180q⁶⁰ + 812q⁶¹ - 1476q⁶² - 825q⁶³ - 242q⁶⁴ + 1106q⁶⁵ + 601q⁶⁶ - 594q⁶⁷ - 329q⁶⁸ - 277q⁶⁹ + 415q⁷⁰ + 283q⁷¹ - 198q⁷² - 63q⁷³ - 141q⁷⁴ + 123q⁷⁵ + 88q⁷⁶ - 66q⁷⁷ + 5q⁷⁸ - 43q⁷⁹ + 30q⁸⁰ + 20q⁸¹ - 19q⁸² + 4q⁸³ - 8q⁸⁴ + 6q⁸⁵ + 3q⁸⁶ - 4q⁸⁷ + q⁸⁸

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 72]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 6, 11, 5], X[8, 3, 9, 4], X[2, 9, 3, 10], > X[16, 8, 17, 7], X[18, 12, 19, 11], X[20, 14, 1, 13], X[12, 20, 13, 19], > X[14, 18, 15, 17], X[6, 16, 7, 15]]
In[3]:=	GaussCode[Knot[10, 72]]
Out[3]=	GaussCode[1, -4, 3, -1, 2, -10, 5, -3, 4, -2, 6, -8, 7, -9, 10, -5, 9, -6, 8, > -7]
In[4]:=	DTCode[Knot[10, 72]]
Out[4]=	DTCode[4, 8, 10, 16, 2, 18, 20, 6, 14, 12]
In[5]:=	br = BR[Knot[10, 72]]
Out[5]=	BR[4, {1, 1, 1, 1, 2, 2, -1, 2, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 72]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 72]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 72]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 72]][t]
Out[10]=	2 9 16 2 3 19 - -- + -- - -- - 16 t + 9 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 72]][z]
Out[11]=	2 4 6 1 + 2 z - 3 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 72]}
In[13]:=	{KnotDet[Knot[10, 72]], KnotSignature[Knot[10, 72]]}
Out[13]=	{73, 4}
In[14]:=	Jones[Knot[10, 72]][q]
Out[14]=	2 3 4 5 6 7 8 9 10 1 - 2 q + 5 q - 8 q + 11 q - 12 q + 12 q - 10 q + 7 q - 4 q + q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 72]}
In[16]:=	A2Invariant[Knot[10, 72]][q]
Out[16]=	4 6 8 10 12 16 18 20 22 28 30 1 + q + 2 q - 2 q + 2 q - 2 q + 2 q - q + 3 q - 2 q - 2 q + q
In[17]:=	HOMFLYPT[Knot[10, 72]][a, z]
Out[17]=	2 2 2 2 4 4 4 4 6 6 -8 2 2 2 z z 3 z 3 z z 2 z 3 z z z z -a + -- - -- + -- + -- + -- - ---- + ---- + -- - ---- - ---- + -- - -- - -- 6 4 2 8 6 4 2 8 6 4 2 6 4 a a a a a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 72]][a, z]
Out[18]=	2 2 2 2 2 -8 2 2 2 z z 3 z z 2 z 6 z 7 z 8 z 5 z -a - -- - -- - -- + -- - -- - --- - -- + ---- + ---- + ---- + ---- + ---- - 6 4 2 9 7 5 3 10 8 6 4 2 a a a a a a a a a a a a 3 3 3 3 4 4 4 4 4 4 3 z 9 z 11 z 5 z z 8 z 11 z 4 z 6 z 4 z > ---- + ---- + ----- + ---- + --- - ---- - ----- - ---- - ---- - ---- + 11 7 5 3 12 10 8 6 4 2 a a a a a a a a a a 5 5 5 5 5 6 6 6 6 6 4 z 7 z 19 z 14 z 6 z 7 z 2 z 8 z 2 z z > ---- - ---- - ----- - ----- - ---- + ---- + ---- - ---- - ---- + -- + 11 9 7 5 3 10 8 6 4 2 a a a a a a a a a a 7 7 7 7 8 8 8 9 9 7 z 9 z 4 z 2 z 4 z 6 z 2 z z z > ---- + ---- + ---- + ---- + ---- + ---- + ---- + -- + -- 9 7 5 3 8 6 4 7 5 a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 72]], Vassiliev[3][Knot[10, 72]]}
Out[19]=	{2, 4}
In[20]:=	Kh[Knot[10, 72]][q, t]
Out[20]=	3 3 5 1 q q 5 7 7 2 9 2 9 3 4 q + 2 q + ---- + - + -- + 5 q t + 3 q t + 6 q t + 5 q t + 6 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 6 q t + 6 q t + 6 q t + 4 q t + 6 q t + 3 q t + 17 6 17 7 19 7 21 8 > 4 q t + q t + 3 q t + q t
In[21]:=	ColouredJones[Knot[10, 72], 2][q]
Out[21]=	-2 2 2 3 4 5 6 7 8 9 q - - + 7 q - 10 q - 4 q + 28 q - 25 q - 22 q + 67 q - 36 q - 59 q + q 10 11 12 13 14 15 16 17 > 108 q - 32 q - 97 q + 128 q - 16 q - 115 q + 117 q + 2 q - 18 19 20 21 22 23 24 25 > 101 q + 80 q + 12 q - 63 q + 37 q + 10 q - 24 q + 10 q + 26 27 28 > 3 q - 4 q + q

The Alternating Knot 1072

The Alternating Knot 10₇₂