10 68

10_67

10_69

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10 68 Quick Notes

10 68 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X12,4,13,3 X20,13,1,14 X16,5,17,6 X8,19,9,20 X18,9,19,10 X10,17,11,18 X14,7,15,8 X6,15,7,16 X2,12,3,11
Gauss code	1, -10, 2, -1, 4, -9, 8, -5, 6, -7, 10, -2, 3, -8, 9, -4, 7, -6, 5, -3
Dowker-Thistlethwaite code	4 12 16 14 18 2 20 6 10 8
Conway Notation	[211,3,3]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-8][-4] Hyperbolic Volume 11.637 A-Polynomial See Data:10 68/A-polynomial

[edit Notes for 10 68's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for 10 68's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$4t^{2}-14t+21-14t^{-1}+4t^{-2}$
Conway polynomial	$4z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 57, 0 }
Jones polynomial	$-q^{3}+3q^{2}-5q+8-9q^{-1}+9q^{-2}-8q^{-3}+7q^{-4}-4q^{-5}+2q^{-6}-q^{-7}$
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^2 a^6-a^6+z^4 a^4+z^2 a^4+a^4+2 z^4 a^2+3 z^2 a^2+a^2+z^4-z^2 a^{-2} }
Kauffman polynomial (db, data sources)	$a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+6a^{4}z^{8}+4a^{2}z^{8}+a^{7}z^{7}+a^{5}z^{7}+7a^{3}z^{7}+7az^{7}-9a^{6}z^{6}-20a^{4}z^{6}-4a^{2}z^{6}+7z^{6}-5a^{7}z^{5}-16a^{5}z^{5}-30a^{3}z^{5}-14az^{5}+5z^{5}a^{-1}+13a^{6}z^{4}+17a^{4}z^{4}-9a^{2}z^{4}+3z^{4}a^{-2}-10z^{4}+8a^{7}z^{3}+23a^{5}z^{3}+27a^{3}z^{3}+8az^{3}-3z^{3}a^{-1}+z^{3}a^{-3}-7a^{6}z^{2}-5a^{4}z^{2}+7a^{2}z^{2}-z^{2}a^{-2}+4z^{2}-4a^{7}z-8a^{5}z-6a^{3}z-2az+a^{6}+a^{4}-a^{2}$
The A2 invariant	$-q^{22}-2q^{16}+2q^{14}+q^{12}+2q^{8}-q^{6}+q^{4}+2q^{-2}-2q^{-4}+q^{-6}+q^{-8}-q^{-10}$
The G2 invariant	$q^{108}-q^{106}+4q^{104}-6q^{102}+6q^{100}-6q^{98}-q^{96}+13q^{94}-24q^{92}+32q^{90}-30q^{88}+12q^{86}+15q^{84}-46q^{82}+60q^{80}-60q^{78}+34q^{76}+4q^{74}-44q^{72}+64q^{70}-62q^{68}+38q^{66}+4q^{64}-38q^{62}+45q^{60}-36q^{58}+8q^{56}+28q^{54}-46q^{52}+53q^{50}-28q^{48}-5q^{46}+50q^{44}-77q^{42}+79q^{40}-53q^{38}+8q^{36}+41q^{34}-73q^{32}+86q^{30}-67q^{28}+26q^{26}+21q^{24}-55q^{22}+56q^{20}-41q^{18}+4q^{16}+28q^{14}-38q^{12}+31q^{10}-8q^{8}-20q^{6}+43q^{4}-47q^{2}+33-7q^{-2}-20q^{-4}+40q^{-6}-43q^{-8}+39q^{-10}-21q^{-12}+5q^{-14}+12q^{-16}-27q^{-18}+28q^{-20}-25q^{-22}+17q^{-24}-7q^{-26}-q^{-28}+8q^{-30}-13q^{-32}+13q^{-34}-10q^{-36}+7q^{-38}-2q^{-40}-q^{-42}+2q^{-44}-4q^{-46}+3q^{-48}-2q^{-50}+q^{-52}$

Further Quantum Invariants[hide]

Further quantum knot invariants for 10_68.

A1 Invariants.

Weight	Invariant
1	$-q^{15}+q^{13}-2q^{11}+3q^{9}-q^{7}+q^{5}-q+3q^{-1}-2q^{-3}+2q^{-5}-q^{-7}$
2	$q^{44}-q^{42}-2q^{40}+4q^{38}-8q^{34}+6q^{32}+6q^{30}-12q^{28}+3q^{26}+12q^{24}-10q^{22}-5q^{20}+11q^{18}-4q^{16}-9q^{14}+6q^{12}+6q^{10}-6q^{8}-2q^{6}+11q^{4}-q^{2}-11+10q^{-2}+4q^{-4}-12q^{-6}+6q^{-8}+4q^{-10}-6q^{-12}+3q^{-14}-2q^{-18}+q^{-20}$
3	$-q^{87}+q^{85}+2q^{83}-5q^{79}-2q^{77}+8q^{75}+8q^{73}-10q^{71}-17q^{69}+6q^{67}+27q^{65}+4q^{63}-33q^{61}-20q^{59}+30q^{57}+37q^{55}-20q^{53}-45q^{51}+q^{49}+52q^{47}+16q^{45}-47q^{43}-32q^{41}+37q^{39}+42q^{37}-25q^{35}-51q^{33}+13q^{31}+53q^{29}-2q^{27}-52q^{25}-11q^{23}+51q^{21}+23q^{19}-41q^{17}-34q^{15}+25q^{13}+42q^{11}-5q^{9}-44q^{7}-18q^{5}+42q^{3}+37q-28q^{-1}-45q^{-3}+17q^{-5}+47q^{-7}-7q^{-9}-37q^{-11}-q^{-13}+25q^{-15}+q^{-17}-14q^{-19}-q^{-21}+8q^{-23}-2q^{-25}-q^{-27}-q^{-33}+2q^{-37}-q^{-39}$
4	$q^{144}-q^{142}-2q^{140}+q^{136}+7q^{134}-8q^{130}-8q^{128}-6q^{126}+22q^{124}+20q^{122}-3q^{120}-28q^{118}-47q^{116}+14q^{114}+54q^{112}+56q^{110}-q^{108}-104q^{106}-71q^{104}+13q^{102}+122q^{100}+125q^{98}-51q^{96}-145q^{94}-147q^{92}+41q^{90}+221q^{88}+133q^{86}-44q^{84}-252q^{82}-171q^{80}+119q^{78}+248q^{76}+176q^{74}-154q^{72}-299q^{70}-101q^{68}+173q^{66}+312q^{64}+44q^{62}-255q^{60}-245q^{58}+27q^{56}+301q^{54}+170q^{52}-147q^{50}-273q^{48}-65q^{46}+242q^{44}+215q^{42}-81q^{40}-273q^{38}-109q^{36}+192q^{34}+247q^{32}-8q^{30}-255q^{28}-182q^{26}+77q^{24}+267q^{22}+148q^{20}-133q^{18}-246q^{16}-150q^{14}+158q^{12}+296q^{10}+115q^{8}-172q^{6}-349q^{4}-81q^{2}+268+307q^{-2}+33q^{-4}-335q^{-6}-243q^{-8}+98q^{-10}+279q^{-12}+159q^{-14}-170q^{-16}-201q^{-18}-22q^{-20}+127q^{-22}+128q^{-24}-52q^{-26}-87q^{-28}-27q^{-30}+30q^{-32}+55q^{-34}-17q^{-36}-19q^{-38}-5q^{-40}+19q^{-44}-8q^{-46}-3q^{-48}-3q^{-52}+6q^{-54}-2q^{-56}+q^{-58}-2q^{-62}+q^{-64}$
5	$-q^{215}+q^{213}+2q^{211}-q^{207}-3q^{205}-5q^{203}+10q^{199}+10q^{197}+3q^{195}-8q^{193}-24q^{191}-23q^{189}+6q^{187}+40q^{185}+50q^{183}+23q^{181}-38q^{179}-94q^{177}-84q^{175}+3q^{173}+116q^{171}+168q^{169}+98q^{167}-76q^{165}-239q^{163}-253q^{161}-63q^{159}+225q^{157}+400q^{155}+300q^{153}-55q^{151}-446q^{149}-569q^{147}-265q^{145}+296q^{143}+723q^{141}+665q^{139}+90q^{137}-651q^{135}-1000q^{133}-610q^{131}+291q^{129}+1074q^{127}+1134q^{125}+316q^{123}-842q^{121}-1452q^{119}-978q^{117}+282q^{115}+1434q^{113}+1554q^{111}+441q^{109}-1088q^{107}-1837q^{105}-1162q^{103}+473q^{101}+1802q^{99}+1708q^{97}+222q^{95}-1474q^{93}-1985q^{91}-855q^{89}+983q^{87}+1988q^{85}+1313q^{83}-464q^{81}-1795q^{79}-1541q^{77}+40q^{75}+1504q^{73}+1573q^{71}+238q^{69}-1233q^{67}-1481q^{65}-353q^{63}+1039q^{61}+1358q^{59}+358q^{57}-944q^{55}-1285q^{53}-357q^{51}+931q^{49}+1305q^{47}+412q^{45}-894q^{43}-1389q^{41}-622q^{39}+743q^{37}+1488q^{35}+965q^{33}-394q^{31}-1462q^{29}-1378q^{27}-197q^{25}+1208q^{23}+1742q^{21}+923q^{19}-683q^{17}-1868q^{15}-1658q^{13}-83q^{11}+1695q^{9}+2208q^{7}+912q^{5}-1198q^{3}-2404q-1630q^{-1}+521q^{-3}+2236q^{-5}+2054q^{-7}+153q^{-9}-1772q^{-11}-2102q^{-13}-671q^{-15}+1186q^{-17}+1855q^{-19}+911q^{-21}-647q^{-23}-1424q^{-25}-904q^{-27}+256q^{-29}+969q^{-31}+737q^{-33}-34q^{-35}-596q^{-37}-516q^{-39}-35q^{-41}+318q^{-43}+308q^{-45}+55q^{-47}-163q^{-49}-174q^{-51}-25q^{-53}+80q^{-55}+71q^{-57}+16q^{-59}-32q^{-61}-35q^{-63}-q^{-65}+19q^{-67}+11q^{-69}-6q^{-71}-8q^{-73}-q^{-75}+5q^{-79}+4q^{-81}-3q^{-83}-4q^{-85}+2q^{-87}-q^{-89}+2q^{-93}-q^{-95}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{22}-2q^{16}+2q^{14}+q^{12}+2q^{8}-q^{6}+q^{4}+2q^{-2}-2q^{-4}+q^{-6}+q^{-8}-q^{-10}$
2,0	$q^{58}-q^{54}-q^{52}+2q^{50}+2q^{48}-4q^{46}-4q^{44}+3q^{42}+5q^{40}-3q^{38}-6q^{36}+4q^{34}+8q^{32}-3q^{30}-7q^{28}+q^{26}+4q^{24}-3q^{22}-6q^{20}+2q^{18}+4q^{16}-q^{14}+3q^{12}+2q^{10}-2q^{8}+3q^{6}+7q^{4}-2q^{2}-6+4q^{-2}+8q^{-4}-6q^{-6}-9q^{-8}+6q^{-10}+6q^{-12}-4q^{-14}-3q^{-16}+2q^{-18}+2q^{-20}-2q^{-22}-q^{-24}+q^{-26}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{46}-q^{44}+2q^{42}+q^{40}-4q^{38}+3q^{36}-2q^{34}-8q^{32}+5q^{30}-2q^{28}-7q^{26}+10q^{24}-5q^{20}+9q^{18}+3q^{16}-4q^{14}+2q^{12}+3q^{10}-6q^{6}+2q^{4}+7q^{2}-8+2q^{-2}+10q^{-4}-9q^{-6}-q^{-8}+8q^{-10}-5q^{-12}-3q^{-14}+5q^{-16}-q^{-18}-2q^{-20}+q^{-22}$
1,0,0	$-q^{29}-q^{25}-2q^{21}+2q^{19}+2q^{15}+2q^{11}+q^{5}+q-q^{-1}+2q^{-3}-2q^{-5}+q^{-7}+q^{-11}-q^{-13}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{46}+q^{44}-4q^{42}+5q^{40}-8q^{38}+11q^{36}-12q^{34}+14q^{32}-13q^{30}+12q^{28}-7q^{26}+2q^{24}+4q^{22}-11q^{20}+17q^{18}-23q^{16}+26q^{14}-26q^{12}+25q^{10}-20q^{8}+16q^{6}-8q^{4}+3q^{2}+4-8q^{-2}+12q^{-4}-13q^{-6}+13q^{-8}-12q^{-10}+9q^{-12}-7q^{-14}+5q^{-16}-3q^{-18}+2q^{-20}-q^{-22}$
1,0	$q^{76}-q^{72}-q^{70}+3q^{68}+3q^{66}-3q^{64}-6q^{62}+8q^{58}+3q^{56}-10q^{54}-10q^{52}+5q^{50}+13q^{48}+q^{46}-15q^{44}-7q^{42}+11q^{40}+13q^{38}-4q^{36}-13q^{34}-q^{32}+11q^{30}+5q^{28}-7q^{26}-4q^{24}+7q^{22}+6q^{20}-5q^{18}-7q^{16}+5q^{14}+10q^{12}-2q^{10}-12q^{8}-2q^{6}+11q^{4}+7q^{2}-9-10q^{-2}+5q^{-4}+14q^{-6}+q^{-8}-11q^{-10}-7q^{-12}+6q^{-14}+10q^{-16}-7q^{-20}-5q^{-22}+2q^{-24}+5q^{-26}+q^{-28}-2q^{-30}-2q^{-32}+q^{-36}$

G2 Invariants.

Weight

Invariant

1,0

$q^{108}-q^{106}+4q^{104}-6q^{102}+6q^{100}-6q^{98}-q^{96}+13q^{94}-24q^{92}+32q^{90}-30q^{88}+12q^{86}+15q^{84}-46q^{82}+60q^{80}-60q^{78}+34q^{76}+4q^{74}-44q^{72}+64q^{70}-62q^{68}+38q^{66}+4q^{64}-38q^{62}+45q^{60}-36q^{58}+8q^{56}+28q^{54}-46q^{52}+53q^{50}-28q^{48}-5q^{46}+50q^{44}-77q^{42}+79q^{40}-53q^{38}+8q^{36}+41q^{34}-73q^{32}+86q^{30}-67q^{28}+26q^{26}+21q^{24}-55q^{22}+56q^{20}-41q^{18}+4q^{16}+28q^{14}-38q^{12}+31q^{10}-8q^{8}-20q^{6}+43q^{4}-47q^{2}+33-7q^{-2}-20q^{-4}+40q^{-6}-43q^{-8}+39q^{-10}-21q^{-12}+5q^{-14}+12q^{-16}-27q^{-18}+28q^{-20}-25q^{-22}+17q^{-24}-7q^{-26}-q^{-28}+8q^{-30}-13q^{-32}+13q^{-34}-10q^{-36}+7q^{-38}-2q^{-40}-q^{-42}+2q^{-44}-4q^{-46}+3q^{-48}-2q^{-50}+q^{-52}$

.

Computer Talk[hide]

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 68"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$4t^{2}-14t+21-14t^{-1}+4t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$4z^{4}+2z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 57, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-q^{3}+3q^{2}-5q+8-9q^{-1}+9q^{-2}-8q^{-3}+7q^{-4}-4q^{-5}+2q^{-6}-q^{-7}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -z^2 a^6-a^6+z^4 a^4+z^2 a^4+a^4+2 z^4 a^2+3 z^2 a^2+a^2+z^4-z^2 a^{-2} }

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+6a^{4}z^{8}+4a^{2}z^{8}+a^{7}z^{7}+a^{5}z^{7}+7a^{3}z^{7}+7az^{7}-9a^{6}z^{6}-20a^{4}z^{6}-4a^{2}z^{6}+7z^{6}-5a^{7}z^{5}-16a^{5}z^{5}-30a^{3}z^{5}-14az^{5}+5z^{5}a^{-1}+13a^{6}z^{4}+17a^{4}z^{4}-9a^{2}z^{4}+3z^{4}a^{-2}-10z^{4}+8a^{7}z^{3}+23a^{5}z^{3}+27a^{3}z^{3}+8az^{3}-3z^{3}a^{-1}+z^{3}a^{-3}-7a^{6}z^{2}-5a^{4}z^{2}+7a^{2}z^{2}-z^{2}a^{-2}+4z^{2}-4a^{7}z-8a^{5}z-6a^{3}z-2az+a^{6}+a^{4}-a^{2}$

Vassiliev invariants

V2 and V3:

(2, -3)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $8$ $-24$ $32$ ${\frac {172}{3}}$ $-{\frac {52}{3}}$ $-192$ $-240$ $-64$ $72$ ${\frac {256}{3}}$ $288$ ${\frac {1376}{3}}$ $-{\frac {416}{3}}$ ${\frac {15031}{15}}$ ${\frac {9196}{15}}$ $-{\frac {20156}{45}}$ $-{\frac {199}{9}}$ $-{\frac {2729}{15}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 10 68. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-7	-6	-5	-4	-3	-2	-1	0	1	2	3	χ
7											1	-1
5										2		2
3									3	1		-2
1								5	2			3
-1							5	4				-1
-3						4	4					0
-5					4	5						1
-7				3	4							-1
-9			1	4								3
-11		1	3									-2
-13		1										1
-15	1											-1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 68]]
Out[2]=	10
In[3]:=	PD[Knot[10, 68]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[20, 13, 1, 14], X[16, 5, 17, 6], X[8, 19, 9, 20], X[18, 9, 19, 10], X[10, 17, 11, 18], X[14, 7, 15, 8], X[6, 15, 7, 16], X[2, 12, 3, 11]]
In[4]:=	GaussCode[Knot[10, 68]]
Out[4]=	GaussCode[1, -10, 2, -1, 4, -9, 8, -5, 6, -7, 10, -2, 3, -8, 9, -4, 7, -6, 5, -3]
In[5]:=	BR[Knot[10, 68]]
Out[5]=	BR[5, {1, 1, -2, 1, -2, -2, -3, 2, 2, -4, 3, -2, -4, -3}]
In[6]:=	alex = Alexander[Knot[10, 68]][t]
Out[6]=	4 14 2 21 + -- - -- - 14 t + 4 t 2 t t
In[7]:=	Conway[Knot[10, 68]][z]
Out[7]=	2 4 1 + 2 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 31], Knot[10, 68]}
In[9]:=	{KnotDet[Knot[10, 68]], KnotSignature[Knot[10, 68]]}
Out[9]=	{57, 0}
In[10]:=	J=Jones[Knot[10, 68]][q]
Out[10]=	-7 2 4 7 8 9 9 2 3 8 - q + -- - -- + -- - -- + -- - - - 5 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[10, 68]}
In[12]:=	A2Invariant[Knot[10, 68]][q]
Out[12]=	-22 2 2 -12 2 -6 -4 2 4 6 8 10 -q - --- + --- + q + -- - q + q + 2 q - 2 q + q + q - q 16 14 8 q q q
In[13]:=	Kauffman[Knot[10, 68]][a, z]
Out[13]=	2 2 4 6 3 5 7 2 z -a + a + a - 2 a z - 6 a z - 8 a z - 4 a z + 4 z - -- + 2 a 3 3 2 2 4 2 6 2 z 3 z 3 3 3 7 a z - 5 a z - 7 a z + -- - ---- + 8 a z + 27 a z + 3 a a 4 5 3 7 3 4 3 z 2 4 4 4 6 4 23 a z + 8 a z - 10 z + ---- - 9 a z + 17 a z + 13 a z + 2 a 5 5 z 5 3 5 5 5 7 5 6 2 6 ---- - 14 a z - 30 a z - 16 a z - 5 a z + 7 z - 4 a z - a 4 6 6 6 7 3 7 5 7 7 7 2 8 20 a z - 9 a z + 7 a z + 7 a z + a z + a z + 4 a z + 4 8 6 8 3 9 5 9 6 a z + 2 a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[10, 68]], Vassiliev[3][Knot[10, 68]]}
Out[14]=	{0, -3}
In[15]:=	Kh[Knot[10, 68]][q, t]
Out[15]=	4 1 1 1 3 1 4 3 - + 5 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 q t q t q t q t q t q t q t 4 4 5 4 4 5 3 3 2 ----- + ----- + ----- + ----- + ---- + --- + 2 q t + 3 q t + q t + 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t 5 2 7 3 2 q t + q t